<!DOCTYPE record>
<record>
<article>
<titex><![CDATA[Power operations in elliptic cohomology   and representations
of loop groups]]></titex>
<tihtml><![CDATA[Power operations in elliptic cohomology  and representations
of loop groups]]></tihtml>
<tiunicode><![CDATA[Power operations in elliptic cohomology   and representations
of loop groups]]></tiunicode>
<tinomath>Power operations in elliptic cohomology and representations of
loop groups</tinomath>
<resauthor><![CDATA[Matthew Ando]]></resauthor>
<author>
<autex>
<fntex><![CDATA[Matthew]]></fntex>
<mntex><![CDATA[]]></mntex>
<lntex><![CDATA[Ando]]></lntex>
</autex>
<auhtml>
<fnhtml><![CDATA[Matthew]]></fnhtml>
<mnhtml><![CDATA[]]></mnhtml>
<lnhtml><![CDATA[Ando]]></lnhtml>
</auhtml>
<auunicode>
<fnuni><![CDATA[Matthew]]></fnuni>
<mnuni><![CDATA[]]></mnuni>
<lnuni><![CDATA[Ando]]></lnuni>
</auunicode>
<auascii>
<fnascii>Matthew</fnascii>
<mnascii></mnascii>
<lnascii>Ando</lnascii>
</auascii>
<afftex><![CDATA[Department of Mathematics, The University of Virginia, Charlottesville,
Virginia 22903]]></afftex>
<affhtml><![CDATA[Department of Mathematics, The University of Virginia, Charlottesville,
Virginia 22903]]></affhtml>
<affunicode><![CDATA[Department of Mathematics, The University of Virginia, Charlottesville,
Virginia 22903]]></affunicode>
<currafftex><![CDATA[Department of Mathematics, The University of Illinois at Urbana-Champaign,
1409 W. Green St., Urbana, Illinois 61801]]></currafftex>
<curraffhtml><![CDATA[Department of Mathematics, The University of Illinois at Urbana-Champaign,
1409 W. Green St., Urbana, Illinois 61801]]></curraffhtml>
<curraffunicode><![CDATA[Department of Mathematics, The University of Illinois at Urbana-Champaign,
1409 W. Green St., Urbana, Illinois 61801]]></curraffunicode>
<curremail><![CDATA[ando@math.jhu.edu]]></curremail>
</author>
<cn></cn>
<abstract>
<abstex><![CDATA[
Part I of this paper describes power operations in elliptic
cohomology in terms of isogenies of the underlying elliptic curve.
Part II discusses a  relationship between equivariant elliptic
cohomology and representations of loop groups.  Part III 
investigates the representation of theoretic considerations which
give
rise to the power operations discussed in Part I.]]></abstex>
<abshtml><![CDATA[<P>
Part I of this paper describes power operations in elliptic
cohomology in terms of isogenies of the underlying elliptic curve.
Part II discusses a  relationship between equivariant elliptic
cohomology and representations of loop groups.  Part III 
investigates the representation of theoretic considerations which
give
rise to the power operations discussed in Part I. 

<P>
]]></abshtml>
<absascii>Part I of this paper describes power operations in elliptic cohomology
in terms of isogenies of the underlying elliptic curve. Part
II discusses a relationship between equivariant elliptic cohomology
and representations of loop groups. Part III investigates the
representation of theoretic considerations which give rise to
the power operations discussed in Part I.</absascii>
</abstract>
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<refhtml><![CDATA[<DL COMPACT><DD>
<P>
<DT><A NAME=Adams%3aBrown><STRONG>[Ada71]</STRONG></A><DD>
J. Frank Adams.
A variant of E. H. Brown's representability
theorem.
<EM>Topology</EM>, 10, 1971.
<A HREF="http://www.ams.org/mathscinet-getitem?mr=44:1018">MR
<STRONG>44:1018</STRONG></A>

<P>
<DT><A NAME=Ando%3aPowerOps><STRONG>[And95]</STRONG></A><DD>

Matthew Ando.
Isogenies of formal group laws and power
operations in the 
cohomology theories <IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="/tran/2000-352-12/S0002-9947-00-02412-0/gif-references0/img1.gif"
 ALT="${E}_{n}$">.
<EM>Duke Math. J.</EM>,
79(2), 1995. <A HREF="http://www.ams.org/mathscinet-getitem?mr=97a:55006">MR
<STRONG>97a:55006</STRONG></A>

<P>
<DT><A NAME=Baker%3aHec><STRONG>[Bak90]</STRONG></A><DD>
Andrew Baker.
Hecke operators as operations in elliptic
cohomology.
<EM>J. Pure and Applied Algebra</EM>,
63:1-11, 1990. <A HREF="http://www.ams.org/mathscinet-getitem?mr=91m:55005">MR
<STRONG>91m:55005</STRONG></A>

<P>
<DT><A NAME=BMMS%3aHin><STRONG>[BMMS86]</STRONG></A><DD>
R. Bruner, J. P. May, J. E. McClure, and M. Steinberger.
<!-- MATH: $H_{\infty}$ -->
<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="/tran/2000-352-12/S0002-9947-00-02412-0/gif-references0/img2.gif"
 ALT="$H_{\infty}$"><EM> ring spectra</EM>,
volume 1176 of 
<EM>Lecture Notes in Mathematics</EM>.
Springer, 1986. <A HREF="http://www.ams.org/mathscinet-getitem?mr=88e:55001">MR
<STRONG>88e:55001</STRONG></A>

<P>
<DT><A NAME=Breen%3aFonctionsTheta><STRONG>[Bre83]</STRONG></A><DD>
Lawrence Breen.
<EM>Fonctions th&#234;ta et
th&#233;or&#232;me du cube</EM>,
volume 980 of 
<EM>  Lecture Notes in Mathematics</EM>.
Springer, 1983. <A HREF="http://www.ams.org/mathscinet-getitem?mr=87d:14036">MR
<STRONG>87d:14036</STRONG></A>

<P>
<DT><A NAME=Brylinski%3aEll><STRONG>[Bry90]</STRONG></A><DD>
Jean-Luc Brylinski.
Representations of loop groups, Dirac
operators on loop space, and
  modular forms.
<EM>Topology</EM>, 29:461-480,
1990. <A HREF="http://www.ams.org/mathscinet-getitem?mr=91:58151">MR
<STRONG>91:58151</STRONG></A>

<P>
<DT><A NAME=BakerWurgler%3aEn><STRONG>[BW89]</STRONG></A><DD>
A. J. Baker and U. W&#252;rgler.
Liftings of formal groups and the Artinian
completion of
  <!-- MATH: $v_{n}^{-1}{B}{P}$ -->
<IMG
 WIDTH="69" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="/tran/2000-352-12/S0002-9947-00-02412-0/gif-references0/img3.gif"
 ALT="$v_{n}^{-1}{B}{P}$">.
<EM>Math. Proc. Cambridge
Phil. Soc.</EM>, 106:511-530, 1989. <A
HREF="http://www.ams.org/mathscinet-getitem?mr=90i:55008">MR
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<P>
</DL>

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<doctext><![CDATA[Introduction Operations arising from isogenies In this paper
we shall consider unstable operations among complex-oriented
cohomology theories which arise in the following situation. Suppose
that E is a commutative, 2 -periodic ring theory with E odd ()
0 ; we shall refer to such theories as homogeneous theories.
A homogeneous theory is automatically complex-orientable, and
E 0 is the ring of formal functions on a formal group over 0E
E 0 () (see section sec:CohDef ). Suppose that E and F are two
such theories, and one has the following algebraic data: enumerate
() a ring homomorphism 0F R; a ring homomorphism 0E R; a homomorphism
of formal groups i . enumerate We denote such data (i,,) . If
the theory E is Landweber exact, the homomorphism i is an isomorphism,
and the map of formal groups is an isomorphism, then it is well-understood
how these data give rise to a stable natural transformation P
: E () F () of ring theories such that align P () and P () align
(see, for example, Miller:Ell ; important issues concerning infinite
complexes have recently been thoroughly investigated by HoveyStrickland
). The first result of this paper is to relax the assumptions
about i and and still obtain an unstable operation. ITheorem
thm:A If E is Landweber exact, and there is an element R which
is not a zero-divisor such that becomes an isomorphism over R
1 , then the data (i,,) give rise to a natural transformation
of ring-valued functors R E 0() R 0(), such that () . ITheorem
In the statement we have used the notation RF 0 (X) YX RF 0 (Y)
where X is a CW complex and Y ranges over the finite subcomplexes
of X ; see section sec:CohDef . Operations in elliptic cohomology
theories Our main application is to elliptic cohomology theories.
An elliptic cohomology theory is a homogeneous theory E , given
with an elliptic curve C over 0E , and an isomorphism of formal
groups (see section sec:EllDef ). Suppose that i : 0E R is 0E
-algebra, A is an abelian group, and A i C is a Drinfel'd A -structure
(see section sec:DrinfeldIsogenies ); we abbreviate such data
as (i,) . Then the quotient i C is an elliptic curve; we use
the notation for the isogeny i C i C . A factorization of (i,)
is a pair (,g) consisting of enumerate () a ring homomorphism
0E R , an isomorphism of formal groups i C g . enumerate From
the factorization (,g) of the Drinfel'd A -structure (i,) , one
obtains a homomorphism of formal groups i g . Using the isomorphism
and Theorem thm:A , one has ICor t-it:InternalFromFactorization
If E is Landweber exact and the order of A is not a zero-divisor
in R , then a factorization (,g) of a Drinfel'd A -structure
gives rise to an operation R E 0() R 0(), whose effect on is
(g ) . ICor The proof of Theorem thm:A and Corollary t-it:InternalFromFactorization
will be given in section sec:prfs-A-B . It is well known how
to produce Drinfel'd A -structures on an elliptic curve or a
formal group; see section sec:DrinfeldIsogenies . A factorization
(,g) can often be found when the elliptic curve C over 0E or
its formal group enjoys universally a property which is retained
by the curve i C over R . As an example, we show ITheorem Theorem
thm:ExistenceOfFactorizations thm:B The elliptic cohomology theory
associated to the Igusa quartic has canonical factorizations
for Drinfel'd isogenies of odd order. ITheorem Applying Corollary
t-it:InternalFromFactorization yields the summands of the Hecke
operator, given as integral unstable operations on the the elliptic
cohomology constructed by Landweber, Ravenel, and Stong. As another
example, we construct K -theoretic operations based on isogenies
of the Tate elliptic curve. Relationship to power operations
The proof of Theorem thm:A uses standard constructions for Landweber-exact
theories, Wilson's techniques for studying unstable operations
Wilson:BP , and Ravenel and Wilson's calculation of Hopf rings
for complex cobordism RavenelWilson:HopfRings as extended by
HopkinsHunton and HuntonTurner:hsrephf . The relationship with
power operations appears when one tries to compute the effect
of the operation on . In the situation of Corollary t-it:InternalFromFactorization
, a coordinate t on gives two functions on i , namely the coordinate
i t and the function g t . By Corollary t-it:InternalFromFactorization
, one always has ()( t) g t, and one might wish to express this
element in terms of the coordinate i t . We give three examples
in section sec:PowerOpsExamples : the theory of Landweber, Ravenel
and Stong, the elliptic cohomology based on the Tate curve, and
the theory E 2 . In each case, there is a coordinate in terms
of which the effect of the operation on is a norm for the isogeny
. In the case of E 2 , we explain how the norm is the shadow
of the H structure on MU . Because of the relationship to the
H structure on MU , we call the operations under study here power
operations, even though we do not show that any elliptic cohomology
theory is an H ring spectrum in the sense of BMMS:Hin . The rest
of the paper follows a lead suggested by analogy with K -theory.
Power operations in K -theory have their origin in symmetries
of the representation theory of compact Lie groups; indeed, any
power operation on K -theory extends to equivariant K -theory,
and the effect of on the representation ring R(G) is precisely
its effect on K G(pt) . Part of the dream about elliptic cohomology
has been that it possesses, for a compact Lie group G , an equivariant
extension which is related to representations of the loop group
LG S 1,G of unbased smooth loops on G Brylinski:Ell,Segal:Ell
, just as equivariant K -theory is related to representations
compact Lie groups. It seems natural to ask what the power operations
in elliptic cohomology mean in the context of loop group representations.
Grojnowski and Ginzburg, Kapranov and Vasserot Grojnowski:Ell,GKV:Ell
have developed a notion of G -equivariant elliptic cohomology
in which the relationship with loop group representations takes
a precise form; this relationship is the topic in the second
part of this paper. It is independent from the first part, aside
from a few well known facts about the Tate curve. Let G be a
compact connected Lie group, T a maximal torus, and W its Weyl
group. Let be the circle group, so that ,T is the group of cocharacters
of T . Let E be an elliptic cohomology theory, and C the associated
elliptic curve. Grojnowski Grojnowski:Ell and Ginzburg, Kapranov
and Vasserot GKV:Ell have explained that the fundamental object
of study for E G is the abelian scheme C together with its W
-action. Grojnowski has constructed a complex equivariant theory
based on this idea. The first elliptic cohomology theory was
discovered by Morava in 1973 (see Morava:FmsOfKThy ): the elliptic
curve is the Tate elliptic curve, . Its formal group is the multiplicative
group, so it is a form of K -theory. We call the theory . Looijenga
Looijenga:RootSystems shows that a simple, simply-connected Lie
group G determines canonically a W -equivariant line bundle (G)
over ; he also calculates its W -invariant sections. The main
point of Part part:ellg is the following. ITheorem thm:C The
character map establishes an isomorphism between the -module
of representations of LG of level k and the -module of W -invariant
global sections of (G) k. ITheorem The precise statement of Theorem
thm:C is slightly more complicated, because of the difficulty
in regarding the Tate curve as a quotient of the multiplicative
group, and so of regarding theta functions as sections of line
bundles. See Corollary t-co:Grojnowski , Theorem thm:FinalForm
and, in the toral case, Theorem C:ETRLT . The main ingredients
in the proof of Theorem thm:C are Kac's character formula and
Looijenga's analysis Looijenga:RootSystems of the line bundle
(G) and its sections. Indeed, Theorem thm:C is essentially Theorem
13.3 of Kac:Book ; although, there one works over , and the geometric
picture is more difficult to discern. The result might have appeared
in Looijenga's paper if Kac's character formula had not been
so new at the time. As it happened, it fell to Grojnowski to
assemble these results and connect them with elliptic cohomology;
I am grateful to him for explaining these ideas to me. I hope
that the present account will make these exciting results accessible
to a wider audience. In part part:POLG we work backwards from
the effect of power operations on to identify operations on representations
of loop groups. It turns out that the extra operations in elliptic
cohomology (beyond those available in K -theory) reflect the
interaction of the tensor powers and the circle group acting
on loop groups by rotation; see Propositions t-pr:isog-and-reps-I
, t-pr:isog-and-reps-II-adams , and t-pr:isog-and-reps-III-mag
. There is undoubtedly more to say on this interaction. If G
is simple, simply connected, and simply laced, then all the irreducible
representations of LG of level n appear as summands in the basic
representation'' of level 1 , pulled back by the n th power map
LG n LG on loops Proposition (9.3.9) PressleySegal:Loo . (In
fact, the same holds when G is a unitary group Frenkel:Rep .)
This pull-back may be viewed as the restriction to the subgroup
of loops fixed by the action of the subgroup of order n of the
group of rotations acting on LG . Thus the symmetry which produces
power operations in elliptic cohomology also generates the representation
theory of certain loop groups, much the same as power operations
in K -theory generate the representation theory of U n from its
representation on n . The decomposition into irreducible summands
of n has been studied extensively by Frenkel Frenkel:Rep , Kac
and Wakimoto KacWak:Bra . The relationship of their results to
the present work remains unclear, and we hope to be able to return
to it. The idea of relating power operations to isogenies, originated
in my dissertation. However, my understanding of these issues
has been greatly influenced by Mike Hopkins and Neil Strickland,
and Part part:POell reflects their tutelage. I learned a lot
from some of Neil's manuscripts, particularly, Formal schemes
and formal groups , and I thank him for making them available
to me. The second part of this paper owes its existence to several
long conversations with Ian Grojnowski and Mikhail Kapranov,
who explained their papers to me and exhibited great patience
with my plodding ways. I am grateful to Farshid Hajir and Amnon
Neeman for conversations about theta functions, and to Haynes
Miller and Jack Morava for conversations and encouragement about
many parts of this project. Finally, I am very grateful to the
referee, who made insightful suggestions, and pointed out important
mistakes in earlier versions of this paper. Like the ones (s)he
caught, the mistakes that remain are mine and mine alone. Power
operations and elliptic cohomology part:POell Complex-orientable
cohomology theories sec:CohDef We construct unstable ring operations,
so if E is a cohomology theory taking values in rings, it is
convenient to use the notation E (X) to refer to the unreduced
cohomology of a space X ; the reduced cohomology will be indicated
by (X) . Definition def:CplxOrientable A ring-valued cohomology
theory E is complex orientable if there is a class x 2() such
that i x 2(1) 2(S 2) E 0 (pt), where S 2 is the inclusion of
the bottom cell, and is the suspension isomorphism. A complex
orientation is a choice of x . Definition Most of the examples
we consider are two-periodic. Definition def:homogeneous A ring
theory E is homogeneous if E odd (pt) 0 , and E 2(pt) contains
a unit of E (pt) . Definition A homogeneous theory is automatically
complex-orientable, as we can easily see by calculating E using
the Atiyah-Hirzebruch spectral sequence. In this case we abbreviate
E(X) for E 0(X) . We shall also occasionally write kE for 0(S
k) E -k (pt) and even E for 0E . Many of our constructions use
the geometry of elliptic curves, which is easiest to describe
in the language of schemes. If E is a homogeneous ring theory,
we use the notation S E for E(pt) ; thus X E(X) is a covariant
functor from spaces to affine schemes over S E . In fact, we
get slightly more, since maps between spaces preserve the augmentation
ideal. Many spaces X of particular interest in the study of complex-oriented
theories have the property that E(X) is complete with respect
to the topology defined by its augmentation ideal. An opposite
category of schemes which remembers this structure will ease
the comparison to elliptic curves: we define the category of
(pointed, affine) formal schemes over S E to be the opposite
of the category of complete augmented E -algebras and continuous
homomorphisms. If R is a complete augmented E -algebra, we denote
the corresponding formal scheme by R ; and if X is a formal scheme
over S E , we denote the corresponding algebra by X . A much
more thorough and detailed treatment of these ideas is given
by Strickland Strickland:FSFG . In fact, most of our examples
will be produced from Landweber's exact functor theorem, so they
really give well defined theories only on finite complexes. If
E is a cohomology theory on finite complexes, we define a functor
on infinite complexes by writing (X) YX E (Y) where Y runs over
the finite subcomplexes. This is not a cohomology theory. However,
E does extend non-canonically to infinite complexes Adams:Brown
. If E' is any such extension, then the natural map E' (X) (Y)
is an isomorphism if, for example, X has cells only in even degrees.
This is the case for X or X , where F is (see section sec:Phantoms
). Similarly, if R is an E -algebra, then we write R(X) YX R(Y)
where Y runs over the finite subcomplexes. A key point is that
completion enters base change and products: the E -algebra R
defines a base change align Formal schemes over S E i Formal
schemes over R given by A R A. align Similarly, the product of
two formal schemes is given by A B A B. In particular, if E is
a homogeneous, complex-orientable cohomology theory, and G E
(), then () G E G E . The multiplication on induces a map of
formal schemes G E G E G E, which, together with the structural
map and the identity section align G E S E G E S E, align gives
G E the structure of a commutative group in the category of formal
schemes over E(pt) . It has more structure, that of a formal
group''. We describe it as follows, to ease the comparison with
elliptic curves (see section sec:EllDef ). Geometrically, the
augmentation ideal 0() is the ideal G E (-e) f G E e f 0 0()
of functions which vanish at the identity, and we have a commutative
diagram equation eq:CPandCotangent xy 0() r d (i) G E (-e) d
2E r 0(S 2) r G E (-e) ( G E (-e)) 2. xy equation The free rank-one
E(pt) -module G E (-e) ( G E (-e)) 2 is the module of sections
of the sheaf E e 1 G E E of one-forms at the origin of G E .
Because of eq:CPandCotangent , we write t e dt for the effect
of the natural map () 2E. Definition def:coordinate A coordinate
on G E is a function t G E (-e) such that e dt trivializes the
sheaf E . Definition A coordinate t gives an isomorphism G E
E. Definition def:formalgroup A (commutative, one-dimensional)
formal group over R or R is a commutative group in the category
of formal schemes, isomorphic as a formal scheme to R. Definition
Remark rem:FormalGroupLaw The multiplication on G E is determined
by the element m t G E G E 0E s,t , which is a formal group law
over 0E , and for which we shall use the notation (G E,t) . Remark
Remark rem:CoordOrient It is important to be clear about the
relationship between a coordinate (Definition def:coordinate
) and an orientation (Definition def:CplxOrientable ). Let t
be a coordinate and let v e dt 2E be the associated one-form:
then the element 2() is a complex orientation. On the other hand,
if x is an orientation and v generates 2E as a 0E module, then
vx is a coordinate. Remark Lemma lem:coordinateorient A coordinate
on G E is equivalent to a pair (x,v) consisting of a complex
orientation and a trivialization of the sheaf E . Lemma Remark
rem:EltOfOrder2 The reader may complain that K -theory is the
only classical example of a homogeneous theory. However, familiar
complex-orientable examples give rise to homogeneous theories
in a natural way: if E is a cohomology theory taking its value
in commutative rings with odd E 0 , then EP (X) u,u -1 E (X)
(with u 2 ) is a homogeneous theory (on finite complexes). This
has the following geometric interpretation: let C R be a pointed
smooth curve over a ring R , and let R C be its identity section.
The sheaf C e 1 C R is invertible and trivializable. The total
space T of the -torsor of trivializations of is (non-canonically)
R u,u -1 . The action grades the ring of functions on T ; the
degree of u is 2 . Remark Example: MP eg:MU This example and
the observation of its utility in the present context are due
to Neil Strickland Strickland:FSFG . Start with the universal
formal group law (G,s) over Lazard's ring L. The differential
e ds trivializes the sheaf G L , so that (G,s,u e ds) over L
u,u -1 is the universal triple (formal group, coordinate, nowhere-vanishing
one-form). If (G,s,) is such a triple over a ring R , we shall
use the notations align L (G,s) R L u,u -1 (G,s,) R align for
the resulting maps. We often omit the group G from the notation
when the context is sufficient. L u,u -1 is the coefficient ring
of the cohomology theory represented by the spectrum MP k 2k
MU. It comes equipped with a map of spectra s MP : () L 2MU MP.
Now suppose that E is a homogeneous ring spectrum, and let i
: ER be an E -algebra. From a map of E -algebras E 0MP R we obtain
an element (E 0s MP ) f of E -modules 0 (),R i G E (-e) which
restricts to a generator of E -modules 0 (S 2),R , which is to
say a trivialization of the sheaf i E. In other words, we obtain
a coordinate on i G E . Lemma lem:coordinateorient together with
the standard argument for MU shows that MP enjoys the following
universal property. Proposition prop:MP Let E be a homogeneous,
complex-orientable ring spectrum. enumerate () If i :E R is a
ring homomorphism, then the natural map E 0MP,R Coordinates on
i G E is an isomorphism. Taking i to be the identity, the natural
map RingSpectra MP,E Coordinates on G E is an isomorphism. enumerate
Proposition Therefore, we use the notation (G MP ,s MP ) for
the universal formal group law. Elliptic cohomology theories
sec:EllDef If C is an elliptic curve over S with identity section
S C, let C,e denote the stalk of germs of functions at the identity,
and let C,e denote its completion at its augmentation ideal.
Any function t C,e such that e dt trivializes the sheaf e C S
E , determines an isomorphism C,e E. We use the notation for
the formal scheme C,e . The group structure on C induces a multiplication
, so is a formal group over S . Definition def:EllipticTheory
An elliptic cohomology theory is a triple (E,C,) consisting of
enumerate () a homogeneous theory E , an elliptic curve C S E
, an isomorphism : of formal groups over S E . enumerate When
the isomorphism is clear, we may omit it from the notation. Definition
Example: Weierstrass eg:Weierstrass The closure of the cubic
W 0: y 2 4 x 3 - g 2 x - g 3 x,y,1 2 is an elliptic curve W over
R 1 6 g 2,g 3,, -1 ( g 2 3-27g 3 2) . Taking as origin the point
0,1,0 at infinity, the function t x y is a coordinate on W .
The pair (W,e dt) is the universal pair (elliptic curve, trivialization
of sheaf of one-forms at origin) over 1 6 -schemes Deligne:Tate
. The functor which associates to a 1 6 -scheme the set of elliptic
curves together with an ordered pair of trivializations of the
sheaf of one-forms at the origin is represented by the ring R
u,u -1 ; the universal triple is (W,e dt,u e dt) . In particular,
the map L u,u -1 (,t,u e dt) R u,u -1 sends u to u . Landweber's
exact functor theorem shows that the functor X MU (X) t e dt
R u,u -1 MP (X) (t,u e dt) R u,u -1 is an elliptic cohomology
theory on finite complexes. Example: Igusa eq:Igusa The closure
I' of the quartic I 0: y 2 1 - 2 x 2 x 4 in 2 is singular at
0,1,0 , but its normalization, which is the closure in 3 of I
0: y 2 1 - 2 x 2 x 4 x,x 2,y,1 3, is an elliptic curve I over
R 1 2 ,,, -1 ( ( 2-)) ; the function x is a coordinate at the
identity 0,0,0,1 I . The map I I' is an isomorphism away from
0,1,0 I' , so I' has canonically the structure of a formal group.
The Igusa quartic enjoys the following universal property. A
triple (C,,P) consisting of an elliptic curve C over a 1 2 -scheme
S , a trivialization of , and a point P of exact order 2 , determines
a map S R together with a map C I' such that is an isomorphism
of formal groups. The function g x is the function on C with
divisor equation eq:IgusasEks 2 -1 (e) - 2 -1 (P), equation normalized
so that is e g dx Igusa . The trivialization e dx identifies
the functor of trivializations of the sheaf I with R u,u -1 ;
the map L u,u -1 (x,u e dx) R u,u -1 sends u to u . Once again,
Landweber's exact functor theorem shows that X MU (X) x e dx
R u,u -1 MP (X) (x,u e dx) R u,u -1 is an elliptic cohomology
theory on finite complexes; it is a 2 -periodicization of the
elliptic cohomology theory of LRS . Example: Tate eq:Tate Consider
the power series align s k (q) n1 n kq n 1-q n , a 4 (q) -5 s
3 (q), a 6 (q) - 5s 3 (q) 7s 5 (q) 12 . align Theorem t-th:Tate-defined
enumerate () The series a 6 (q) as well as a 4 (q) has coefficients
in . The Weierstrass cubic equation eq:Tate-Weierstrass y 2 x
y x 3 a 4 (q)x a 6 (q) equation over has discriminant equation
eq:Tate-discr (q) q n1 (1 - q n) 24 . equation enumerate Theorem
proof See, for example, p. 410 Silverman:EllipticCurvesII . proof
The Tate curve, , is the curve over given by eq:Tate-Weierstrass
. It is a pointed curve of genus 1 over . The point is located
in the smooth part, which is a one-dimensional abelian group,
and there is an isomorphism equation eq:GmfandTatef equation
of formal groups (see VII, 1.16 DeligneRapoport ). The coordinate
x x y gives a trivialization of the sheaf , so the ring of functions
on the -torsor of trivializations of is canonically isomorphic
to u,u -1 . It follows that the functor align X MU (X) x e dx
u,u -1 MP (X) (x ,u e dx ) u,u -1 align is a cohomology theory
on finite complexes. Because its formal group is multiplicative,
it is a form of K -theory Morava:FmsOfKThy ; we call it . The
formula eq:Tate-discr for the discriminant implies that over
, the Tate curve is an elliptic curve. It is modeled on the multiplicative
parameterization of elliptic curves over : if q is a complex
number with 0 q 1 , then q is an elliptic curve which fits into
an exact sequence 1 q q 1; we obtain in this way a family of
elliptic curves q D where () q q 1 . Indeed, for q() , the series
a 4 (q) and a 6 (q) converge to complex numbers; let E q denote
the resulting elliptic curve over . The series align X (z,q)
n q nz (1-q nz) 2 - 2 s 1 (q), Y (z,q) n (q nu) 2 (1-q nu) 3
s 1 (q), align define meromorphic functions of z for z . The
function align (z) X (z,q):Y (z,q):1 ,z q , (q r) e 0:1:0 align
defines an analytic isomorphism q E q; see p. 410 Silverman:EllipticCurvesII
. Alas, q () is not a -valued point of . However, if F is a non-archimedean
field, let (F) qF q 1 . The function align cts ,F (F) g g (q)
align is an isomorphism of sets. Theorem t:TateCurveII For any
continuous homomorphism F, the procedure above gives an isomorphism
of groups F g (q) g (F). Theorem proof See, for example, p. 423
Silverman:EllipticCurvesII . proof Using the function z 1 - x
, we define another coordinate (z,q) on the formal group by the
formula equation eq:sigma (z,q) (1-z) k1 (1-q kz)(1-q kz -1 ).
equation It does not extend to a function on the Tate curve.
Instead, it is a variant of the Weierstrass function: if F is
either or a p -adic field, then for q(F) , defines a function
on F , which has zeroes of first order at the points q and which
satisfies the equation (qz,q) -1z(z,q). Its relevance here is
that (-z,q) is the character of the basic representation of the
loop group of a circle; see section sec:LT . We recall a model
for the torsion of the Tate curve from 8 KatzMazur:Ari . Let
A q,q -1 , and let N be the A -scheme which is the disjoint union
N i T i N . Here T i N is the scheme T i N ( A t (t N -q i) ).
If R is an A -algebra, then T i N (R) rR r N q i . The scheme
N is a group scheme via the multiplication align T i N T j N
T i j N (r,s) rs. align The constant group is a subgroup of N
by k q k Nk N ; we denote this subgroup q . Let T N be the quotient
N q ; as a scheme it is isomorphic to the subscheme 0iN-1 T i
N of N . As a group it fits into a short exact sequence of group
schemes align N T 0 N T N 1N T i . align Here N denotes the group
scheme of N th roots of unity. The maps align T i N T ik Nk r
r align for various i, k, and N induce homomorphisms T N T Nk
. Let T be the resulting ind-groupscheme; it fits into a short
exact sequence 1T 0 of ind-groupschemes. Theorem thm:TateCurveDivisGp
Over there is a canonical isomorphism of ind-groupschemes T which
is compatible with the isomorphisms eq:GmfandTatef and T 0. Theorem
proof See 8 KatzMazur:Ari . proof If R is a complete topological
ring, then a continuous homomorphism R is determined by f (q)
, and we often follow the convention of writing (f (q)) and T
(f (q)) N for f and f T N , etc. Example: E 2 sec:ExE2isElliptic
Let C 0 be a supersingular elliptic curve over a perfect field
k of characteristic p 0 , and let E (C 0) be the Lubin-Tate ring
of deformations of its formal group C 0 . The Serre-Tate Theorem
(see, for example, Katz:SerreTate ) gives a canonical lift C
of C 0 to E (C 0) , such that is the universal deformation of
C 0 . A choice of coordinate t on gives ring homomorphism 0MP
(,t) E (C 0), which satisfies the hypotheses of Landweber's exact
functor theorem, so the functor E (C 0) () MP () (,t) E (C 0)
is a cohomology theory on finite complexes. Indeed a choice of
p -typical coordinate t determines an isomorphism E (C 0) (pt)
k u 1 u,u -1 , exhibiting E (C 0) as a cousin of the completed''
2 -periodic E 2 of JohnsonWilson:En,BakerWurgler:En . More generally,
if G 0 is a formal group over k of height n , let E (G 0) be
the Lubin-Tate ring of deformations of G 0 . A choice of coordinate
t on the universal lift G defines a cohomology theory E (G 0)
, which is a cousin of the completed 2 -periodic E n . Uniqueness
and integrality using Hopf rings sec:Phantoms The preceding section
dealt with cohomology theories defined on finite CW complexes.
In this and later sections we shall discuss unstable natural
transformations between such theories. Since the operations we
construct are multiplicative, we treat these cohomology theories
as functors from finite complexes to rings. Such theories and
operations can be extended to infinite complexes, although in
general not canonically. However, with the Hopf ring techniques
initiated by RavenelWilson:HopfRings,Wilson:BP , it is possible
to remove much of the ambiguity with regard to infinite complexes,
at least unstably. The following propositions cover the situations
which arise in this paper. Suppose that E and F are homogeneous
theories, and that F 0 has a Kunneth isomorphism (the space appears
not to be uniquely defined; Corollary t-co:UniquenessOfInfLoopSpace
shows that this is not the case). Choosing coordinates sE 0 and
t F 0 gives classes i F 0 for i1 and a map . We then have classes
a and b i in F 0 for a 0E and i1 , defined by subequations eq:HopfRingGenerators
align 0 F 0 eq:hurewicz a a , F 0 F 0s F 0 i b i align subequations
(the map eq:hurewicz is the Hurewicz homomorphism for F ). The
methods of Wilson:BP apply when E 0 is the quotient of the free
Hopf ring on the generators eq:HopfRingGenerators by the relation
of Theorem 3.8 in RavenelWilson:HopfRings (in fact, we shall
use here only the fact that the classes eq:HopfRingGenerators
generate the Hopf ring). If E is such that F 0 is of this form
for all homogeneous F , then we say that E is . Hopkins and Hunton
HopkinsHunton show that E is if it is Landweber exact, and 0E
is countable over a subring of . More recently, Goerss and independently
Hunton and Turner HuntonTurner:hsrephf have removed the hypothesis
of countability. As an example of the utility of this description,
we first observe the following uniqueness. Proposition prop:NoPhantoms
Let E and F be two homogeneous, complex-orientable ring spectra,
and suppose that E is . Suppose that a and b are two maps E a,b
F, such that af and b f are homotopic for any map X E out of
a finite complex X . Then a and b are homotopic. Proposition
proof If E is , then there are isomorphisms align E, F F 0( E)
F F 0 E,F . align Moreover, F 0 E is spanned as an F -module
by classes which are in the F -homology image of maps to E from
finite products of spheres and finite complex projective spaces.
proof Corollary t-co:UniquenessOfInfLoopSpace For example, the
proposition shows that an isomorphism E () c() F () of functors
on the category of finite complexes gives rise to a well defined
homotopy equivalence E F. Corollary Integrality of unstable operations
We recall and slightly generalize Wilson's method in 11 Wilson:BP
for identifying integral unstable operations. Suppose that E
and F are homogeneous theories, E is , and S is an F -algebra.
The generalization involves the study of natural transformations
of ring-valued functors equation eq:ring-functor-op E () S F
(). equation Of course S F is not, in general, a cohomology theory,
but it is still true that an additive natural transformation
E () S F () is given by a primitive element gPS F (E) . The assumption
that E is implies that E is weakly equivalent to a CW complex
C with cells concentrated in even degrees. For any finite subcomplex
Y of C the natural map SF 0Y F F 0Y,F S is an isomorphism. Using
these F 0Y to filter F 0E , it follows that there are isomorphisms
equation eq:coh-and-hom split S F 0E F F 0E,F S , PS F 0E F QF
0E,F S, split equation where QF 0E denotes the F -module of indecomposables
of F 0E . Now suppose that i : FR is an F -algebra, and R is
not a zero-divisor, so that RR 1 is injective. Let us use the
notations align R () R F () , T () R 1 F () align for the indicated
ring-valued functors. Given a natural transformation, we suppose
E () T (), and so a well defined element of PT 0 F QF 0,F R 1
. Proposition Wilson t-pr:Wilsons-method The operation g refines
to a natural transformation E () R () if and only if g () factors
through R () and g () factors through R () . Proposition proof
The only if direction is obvious. So suppose that g () and g
() factor through R () . The hypotheses give that in the diagram
CD PR 0() F QF 0,F R VVV VVV PT 0() F QF 0,F R 1 , CD the horizontal
arrows are isomorphisms and the vertical arrows are monomorphisms.
Choose coordinates s and t on and , and let a and b i be defined
as eq:HopfRingGenerators . The F -module QF 0 is spanned as an
F -module by the image from F 0 of elements of the form x a b
1 i 1 b 2 i 2 b r i r . If g () (s) i1 u i t i, then it is not
hard to check that align ( a ) g () (a) , (b k) u k , (x) (g
() (a)) k 1 r u k i k . align It follows that g induces an operation
E () R() as promised. proof The same methods give the following
result. Proposition t-pr:WilsonsII If E is , then two ring operations
from E 0() to S F 0() coincide if they coincide on a point and
. Proposition Drinfel'd isogenies sec:DrinfeldIsogenies Level
structures and isogenies It will be helpful to recall some facts
about level structures and isogenies of formal groups and elliptic
curves. We start with a formal group or elliptic curve C over
a scheme S , and an abelian group A . If C is a formal group,
then we suppose that S is the formal spectrum of a complete local
ring. Definition See KatzMazur:Ari def:Drinfeld-level-structure
A Drinfel'd A structure on C A C is a homomorphism of groups
A C(S), such that the Cartier divisor a A (a) is a subgroup scheme
of C . Definition If : AC is a Drinfel'd A structure, then, in
particular, the subgroup-scheme is finite and flat over S . In
that case, the quotient of C by is a formal group or elliptic
curve, according to the type of C (see 12 Mumford:AbelianVarieties
, Ch. III, 2 DemazureGabriel , Lubin:Fin , and Strickland:FiniteSubgps
). Definition def:DrinfeldIsogeny A Drinfel'd isogeny is a sequence
A C D consisting of an isogeny and a Drinfel'd level structure
such that . Definition Cohomology theories with level structures
sec:Levelstructures It is easy to produce Drinfel'd isogenies
on elliptic curves or formal groups. Proposition prop:DrinfeldModuliSpaces
If C is an elliptic curve or formal group over a ring E , and
A is a finite abelian group, then there is a universal pair (i,)
consisting of an E -algebra E R (in case C is a formal group,
i is a continuous map of complete local rings) and a Drinfel'd
level structure A i C. The homomorphism i is flat. Proposition
proof If C is an elliptic curve and A is cyclic or isomorphic
to ( N) 2 , this is a result of Drinfeld:Ell and Theorem 5.1.1
KatzMazur:Ari . For general A and C a formal group, see Strickland:FiniteSubgps
. The case of a formal group is in turn the difficult input (axiom
4B) of Theorem 5.2.1 in KatzMazur:Ari , which gives the case
of general A and an elliptic curve C . proof In particular, if
(E,C,) is an elliptic cohomology theory, then the functor X F
(X) R E (X) is a cohomology theory on finite complexes, equipped
with a map E () F () of ring theories. The norm of a coordinate
sec:WhenIsNormCoordinate Let A C D be a Drinfel'd isogeny over
S . Let t be a function on C for which t((a)) for a A , and define
equation eq:norm N t a A (a) t, equation where (a) : C C is translated
by (a) . The standard construction of the quotient D ( 12 Mumford:AbelianVarieties
and Ch. III, 2 DemazureGabriel ), together with Proposition 1.10.6
in KatzMazur:Ari , shows that N t descends to a function t on
D . Now suppose, in addition, that t is a coordinate on C . In
section sec:HeckeIgusa it will be useful to know when t is a
coordinate on D . Proposition prop:NormIsCoordinate t is a coordinate
on D if and only if the following condition is satisfied at each
geometric point x of S : for all a A , we have t x((a)) 0 if
and only if (a) e x . Proposition proof It suffices to check
that t is a coordinate on the fiber at each geometric point of
S , so by base change we are reduced to the case that S is the
spectrum of an algebraically closed field k . If the characteristic
of k is zero, then is unramified so C D is etale. Then t is a
coordinate on D if and only if N t is a coordinate on C . The
equation align eq:NormDifferential e dN t e dt, where 0a A t((a)),
align shows that t is a coordinate if and only if t((a)) 0 for
0 a A . If the characteristic of k is p 0 , then ramifies, say
to degree q p r . In this case factors as C Fr C (q) D, where
Fr is the q -power Frobenius and u is etale II, 2 Silverman:EllipticCurves
. t is a coordinate on D if and only if u t is a coordinate on
C (q) . Let A f A be the subgroup A f a A (a) e . Let A e A be
a splitting of A A A f ; and write for the composition A e C
C (q) . Then A e C (q) D is a Drinfel'd isogeny. Let z be the
coordinate on C (q) such that (Fr) z t q . We have align t a
A (a) t a A e b A f (a) (b) t a A e (a) ( b A f (b) t) a A e
(a) (t q) Fr ( a A e (a) z), so u t N z. align For eq:NormDifferential
shows that t is a coordinate on D if and only if t((a)) 0 for
0 a A e. proof Unstable operations from homomorphisms of formal
groups sec:prfs-A-B Suppose that F is a homogeneous theory; i
: 0F R is a 0F -algebra; and i G F H is a homomorphism of formal
groups, which becomes an isomorphism after inverting an element
R which is not a zero-divisor. Let u be a coordinate on H . Proposition
t-pr:UnstableMUoperation There is an unstable natural transformation
MP 0() Q() R F 0() of ring-valued functors, such that Q(pt) classifies
the formal group law (H,u) , and Q()(s MP ) u R 0 (). Proposition
proof The coordinate u gives a function u on i , which is a coordinate
over R 1 . By Proposition prop:MP it is classified by a ring
homomorphism f F 0MP,R 1 . Using the fact that MP is and the
isomorphisms eq:coh-and-hom , we obtain from f an element of
align F QF 0MP,R 1 F QF 0MP,F 1 PR 1 0 (MP). align Let Q denote
the resulting natural transformation (of ring-valued functors)
MP 0 () R 1 0 (). By construction we have Q () (s MP ) u , so
Q (pt) classifies the formal group law (i G F, u) (H,u) . Proposition
t-pr:Wilsons-method now implies the result. proof Proof of Theorem
thm:A We are now prepared to give a proof of Theorem thm:A .
Thus suppose that E and F are homogeneous theories; E is Landweber-exact;
align 0F R 0E R align are ring homomorphisms, and that is a homomorphism
of formal groups i G F G E which becomes an isomorphism when
R is inverted. The element is further required not to be a zero-divisor.
Choose a coordinate u on . By Proposition prop:MP , this gives
a ring operation MP 0 () E 0 () such that t (pt) classifies the
group law (,u) . By Proposition t-pr:UnstableMUoperation it also
gives a ring operation MP 0 () R 0 () such that Q (pt) classifies
(, u) ; in other words, Q (pt) t (pt). It follows that the diagram
xy 4pc MP 0() r Q d t (pt) R 0() E 0(pt) t (pt) MP 0() r - ru
Q E 0() -- u xy commutes; since E is Landweber-exact, the natural
map E 0 (pt) t (pt) MP 0 () E 0 () is an isomorphism, which fills
in the dotted arrow. By construction, the effect of on u R E
0 () is given by () (u) Q () (s MP ) u, so equation eq:EffectOfPsiOnCoord
() . equation Equation eq:EffectOfPsiOnCoord shows that the operation
is independent of the choice of coordinate u . More precisely,
let s be another coordinate on , and denote by s and t the two
elements of R 0( E) produced by the proof of Theorem thm:A .
I claim that s t . Proposition t-pr:WilsonsII reduces the problem
to comparing the operations on a point and . They coincide on
a point since s(pt) t(pt). They coincide on since s() t(). Proof
of Corollary t-it:InternalFromFactorization Given Theorem thm:A
and the remarks preceding the statement of Corollary t-it:InternalFromFactorization
, all that remains is to observe that if N A , then the dual
isogeny gives a factorization xy i C r - dr N i C d i C , xy
and that N induces an isomorphism of formal groups over R 1N
. Examples sec:PowerOpsExamples In this section we construct
isogeny data, and so operations, for the theories based on the
Igusa quartic, E 2 , and the Tate curve. In the first two cases,
the method is to observe that the elliptic curve C S E is universal
for some property, and then to find an instance of that property
on the quotient elliptic curve D S F , which is classified by
a map S F S E. In all three cases, there is a coordinate on the
elliptic curve C in terms of which the effect of () is given
by a norm. The appearance of this norm was foreshadowed by formulas
for the H structure on MP , in honor of which we call the operations
power operations. Igusa sec:HeckeIgusa Let (E,I) be the elliptic
cohomology based on the Igusa quartic (section eg:Weierstrass
), let N be an odd integer, and let S F F S E be the scheme of
Drinfel'd ( N) 2 -structures on I , with universal example (
N) 2 i I. For each subgroup A ( N) 2 we obtain by restriction
a level- A structure A which fits into a Drinfel'd isogeny A
A i I A i I A i I A . Write Ax for the function A i x on i I
A . The first point is Lemma lem:IgusaDescends Ax is a coordinate
on i I A . Lemma proof Proposition prop:NormIsCoordinate applies.
The function x is constructed to have divisor 2 -1 (e) - 2 -1
(P), where P is a point of strict order 2 ; and this feature
of x is preserved under base change. If the order of A is odd,
then x never has a pole at (a) for a A , and it vanishes at (a)
if and only if (a) e . proof Let A and g A be the maps align
S F A S E i I A g A A I' align associated to the triple (i I
A,e d Ax, A(P)) by the universal property of Igusa's quartic
(see section sec:EllDef ). By construction we have Theorem Theorem
thm:B thm:ExistenceOfFactorizations The pair ( A, A) is a factorization
of (i, A). Theorem Theorems thm:A and thm:ExistenceOfFactorizations
together construct an operation A for the elliptic cohomology
given by the Igusa quartic. In fact, one can be quite explicit
about the effect of A() in this case. Proposition prop:EffectOfPsiIsNorm
A()(x) N Ai(x) . Proposition proof By construction we have align
A A A x A()(x) and A Ax N Ai(x) . align We must show that g A
A x Ax . Now g A x is the function on i I A satisfying the equations
align g A x 2 -1 (e) - 2 -1 ( A(P)), e dg A x e d Ax. align It
is easy to check that Ax also satisfies these two equations.
proof Corollary For N odd there is an additive natural transformation
T N A ( N) 2 A N A from E 0() to E 0() . Corollary proof The
sum A ( N) 2 A N A clearly gives an additive operation E 0()
T N F 0(), or equivalently (since E is ) an element of E E 0(
E),F . What remains is to show that this element is in the subset
E E 0( E),E E E 0( E),F . Indeed it suffices to show that it
lies in the subset E E 0( E),E 1N E E 0( E),F 1N . Now Baker's
argument Baker:Hec applies; the point is that T N(pt) is just
N times the classical Hecke operator. proof Power operations
sec:PowerOps Suppose that C 0 is a formal group of height n over
a finite field k of characteristic p 0 , and let E (C 0) be the
resulting cohomology theory as in section sec:EllDef . Then C
E (C 0) is the Lubin-Tate deformation of C 0 . Lubin and Tate
LubinTate:Com show that over the ring of integers of the algebraic
closure of the fraction field of E (C 0) we have (( p ) n). Let
A be a finite subgroup of ( p ) n of order q , and let i : E
(C 0) E (C 0,A) be a universal complete local E (C 0) -algebra
equipped with a level- A structure A i C i C A as in Proposition
prop:DrinfeldModuliSpaces . Proposition prop:DrinfeldModuliSpaces
implies that i is flat; let E (C 0,A) () E (C 0,A)E (C 0) ()
be the resulting cohomology theory. Since C 0 (k) 0 , the homomorphism
induces the zero homomorphism over k . It follows that there
is an isomorphism (i C A) 0 C (q) 0 of formal groups over k ,
which identifies the isogeny 0 with the q -power Frobenius. It
follows that i C A is a deformation of C 0 (q) . Letting H over
E (C 0 (q) ) be the universal deformation of the formal group
C 0 (q) , it follows that there is a map of local rings E (C
0 (q) ) () E (C 0,A) () and an isomorphism of formal groups i
C A H. The data (i,g,) produces, using Theorem thm:A , a ring
operation E (C 0 (q) ) 0() E (C 0,A) 0(). Moreover, if k so that
C 0 C 0 (q) , then we have E (C 0) () E (C 0 (q) ) (), and so
an operation E (C 0) () E (C 0,A) 0(). It is in this case that
the relationship with the H structure on MP is easiest to explain.
Let t be any coordinate on C such that t((a)) for a A . C 0(k)
is the one-element group, so the condition of Proposition prop:NormIsCoordinate
is automatically satisfied. The resulting coordinate t on i C
A yields an operation MP 0() E (C 0,A) 0() by Proposition t-pr:UnstableMUoperation
. Another construction of this operations goes as follows Ando:PowerOps
. The H structure on MP gives an operation MP 0() MP 0(BA ) (where
A A, ), and the character theory of HKR gives an operation E
0(BA ) HKR E (C 0,A) 0 (). The composite MP 0() MP 0(BA ) E 0(BA
) HKR E (C 0,A) 0() is Q . In Ando:PowerOps we showed that one
can choose the coordinate t so that the diagram xy MP 0() d t
24pt,0pt dr Q E (C 0) 0() r E (C 0,A) 0() xy commutes, and so
()(t) N Ai(t), as in the case of the Igusa quartic. It is in
honor of this relationship with the H structure on MP that we
call the operations based on isogenies power operations. The
Tate curve sec:HeckeTate There are two basic families of isogenies
out of the Tate curve. If a is a positive integer, then there
is an isogeny (q) a (q a) with kernel a , modeled on align F
q a F q a z z a. align Over q 1 b there is an isogeny (q) b (q
1 b ) with kernel 1b , modeled on the projection align K q b
K q 1b z z q 1b . align More generally, let D (a,b) be the A
-algebra D (a,b) A r (r b q a). There are two ring homomorphisms
D (a,b), namely, i (q) q and (q) r , and an isogeny g a,b : i
(q) a (q a) b (r) . The kernel of g a,b is the D (a,b) -scheme
T a,b z T(q) ab z a r i for some i . It is a disjoint union of
components T i a,b T a,b T ia ab for i 0,,b-1 , and fits into
a short exact sequence of the form a T a,b 1b . The following
is easy to check using formulae which are available, for example,
from Silverman:EllipticCurvesII . Lemma lem:CoordinatesOnTateCurveUnderIsogeny
The diagram equation eq:g-ab-on-fg CD i a,b V i x VV VV x V a
CD equation commutes. Lemma Applying Theorem thm:A , we have
Corollary cor:OperationsInKTate For a,b1 there is a ring operation
D (a,b)() a,b D (a,b) () such that, considering x as an element
of () , equation eq:PsiOnTate a,b ()( x ) a i x . equation In
terms of the function z 1-x , we have equation eq:PsiOnTate-z
a,b () ( z) (i z) a. equation Corollary Although the coordinate
eq:sigma is not a function on the Tate curve, so Proposition
prop:NormIsCoordinate is not directly relevant, it is still true
that the effect of a,b on is given by a suitable norm. We treat
the cases 1,n and n,1 . By equation eq:PsiOnTate-z , we have
1,n ()((z,q)) (z,r), while we check directly that equation eq:PsiOnSigma
N g 1,n (z,q) j 0 n-1 (zr j,q) (z,r). equation Similarly, equation
eq:PsinOnSigma n,1 () ((z,q)) (z n,q n) j 0 n-1 (z j,q) N g n,1
(z,q), equation where is a primitive n th root of 1 . Remark
rem:SigmaAnd0 In fact, is a section of the line bundle associated
to the divisor (e) over ; see Remark rem:L-G-generally and the
discussion of the basic representation of L in section sec:LT
. The formulae here describe the behavior of this line bundle
under the norm maps defined by various isogenies. Remark Remark
rem:Drinfeld-Level-structure Let denote a primitive b th root
of unity. If R is a -algebra and rR satisfies r b q a, then the
R -valued points of D (a,b) are just (D (a,b)) (R) r j 0j b-1
; the resulting isogenies are of the form (q) (q a) (r j). We
use the notation a,b, j for the associated operation. For consistency
with the discussion in Part part:POLG , it will be convenient
to study D (1,b) under the ring homomorphism align s b q q b.
align This map extends over i :D (1,b) by s b (r) q. Indeed for
b 1 there is an extension of s b : given by s b, (r) q. Under
this specialization, the isogeny g 1,b becomes g 1,b, : (q b)
(q) Remark Elliptic cohomology and loop groups part:ellg Notation
We will make use of the following notation and conventions. align
: the multiplicative group , : the circle group, written multiplicatively
, T : a compact torus , : T, , : ,T . align We honor the tradition
of identifying and with lattices in the Lie algebra of T and
its -linear dual, by writing the group structure additively.
More generally, if L is a lattice (finitely generated free abelian
group) and L : L, is its dual, then we frequently make use of
the canonical isomorphisms align L , L , L L, . align The point
is that the right-hand side in each case is a constant group
scheme. If A is an abelian group written additively, and M is
an abelian group written multiplicatively, we shall observe the
tradition of writing m a for the element a m of A M . Elaborating
on these traditions, if n L and z L , we write z n for the resulting
element of , etc. Equivariant elliptic cohomology sec:EquivariantEll
If G is a compact Lie group, denote by G() K G() the spectrum
of G -equivariant K -theory; it is a functor to G G(pt) RG -schemes.
The isomorphism K(B) is the completion of the isomorphism R .
More generally, a celebrated theorem of Atiyah-Segal says that
if T is a compact torus, then K(BT) is a completion of equation
eq:KT T , equation and if G is a connected compact Lie group,
T a maximal torus, and W its Weyl group, then K(BG) is a completion
of equation eq:KG G RG () W. equation If (E,C,) is an elliptic
spectrum, then equation eq:ECPCf E(B) equation and it has always
seemed desirable to construct an equivariant theory for which
C, whose completion is eq:ECPCf . The difficulty is that the
elliptic curve C has no non-constant holomorphic functions. Grojnowski
and Ginzburg, Kapranov and Vasserot address this problem by ignoring
it: should be a functor from -spaces to sheaves over C . Generalizing
eq:KG , G should be a functor from G -spaces to W -equivariant
sheaves over C. Grojnowski has constructed such a theory, based
on the elliptic curve ( ) over the upper half-plane. Here we
shall study the equivariant extension of . It appears likely
that Grojnowski's method carries over to this case, although
we shall not address that problem here: we will be concerned
only with ( ) G ( ) G(pt) (), with its W -action. From now on
we write . It turns out that characters of representations of
the loop group LG are precisely the invariant sections of a canonical
W -equivariant line bundle over G . We reiterate that this is
really a simple consequence of results of Kac and Looijenga,
communicated to the author by Grojnowski; see the remarks at
the end of the introduction. Starting from a recollection about
representations from PressleySegal:Loo , we assemble results
from Looijenga:RootSystems and Kac:Book to arrive at Theorem
thm:C . This much detail provides a foundation for the discussion
in Part part:POLG . Representations of loop groups s:LoopGroups
We denote by LG S 1,G the group of smooth unbased loops in a
compact connected Lie group G . Once we settle on an appropriate
notion of representation for loop groups, the representation
theory bears strong resemblance to the representation theory
of compact Lie groups. We indicate some highlights, which we
learned from PressleySegal:Loo . Before describing the representations,
note that: enumerate () The group G includes as the constant
loops in LG . The circle group acts on LG by rotating loops.
enumerate We write for the circle group acting in this way. Fix
once and for all a generator q of rot , so that R () q,q -1 .
For a we shall denote by LG a the semi-direct product formed
from the action rot (LG). The representations of LG studied in
PressleySegal:Loo are projective representations of a definite
level: let E LG be a central extension of LG by the circle group
; we write for the central circle. Definition def:admiss-extn
A G - split central extension E is a central extension of LG
by , together with enumerate () a splitting E G of the central
extension over the constant loops; and an action of on E lifting
the action on LG . enumerate Definition If E is G -split, then
we may form the semi-direct product Ea . Given a continuous action
of E a on a topological vector space V , the subspace of vectors
of energy k '' is the vector space V (k) v V acts by the character
q k . Definition def:Representation Let E be a G -split central
extension of LG . A representation of LG of level E and type
a is a unitary representation of E a on a separable Hilbert space
V , such that enumerate () for k sufficiently small, V (k) 0
; it:bne each V (k) is finite dimensional; it:finite-V-k the
center of the extension acts by the standard character of . it:center-acts
enumerate Definition Remark rem:Why-the-E The definition of representation
really gives a particular class of representations of E . However,
if G is simple and simply connected, a torus, or the unitary
group U r , there is a natural choice of central extension E
, denoted LG in PressleySegal:Loo ; see also sections sec:simple-simply-connected
and sec:LT of the present paper. The level of generality here
is intended to indicate more clearly the connection with theta
functions in section sec:relationship-to-theta . Remark Remark
rem:level The restriction it:center-acts is not standard but
also not serious. What in PressleySegal:Loo might be called a
representation of E of level h would here be called a representation
of level h E . Remark Remark rem:bne The condition it:bne of
Definition def:Representation says that V has bounded negative
energy''. In PressleySegal:Loo one restricts to V such that V
(k) 0 for k 0 . The restriction is not serious since if V has
bounded negative energy, then there is a k such that q kV has
positive energy. Remark Definition def:EssentialEquiv An essential
equivalence Chapter 9.3 PressleySegal:Loo from V to W is a continuous
equivariant inclusion VW with dense image. Definition With these
restrictions, the representation theory of LG resembles closely
the representation theory of a compact Lie group. enumerate ()
Every representation is essentially equivalent to a discrete
direct sum of irreducible representations. We shall say that
two representations V and W are equivalent if there are a discrete
direct sum of irreducible representations D I V and essential
equivalences V D W. i:fchar The equivalence class of a representation
is determined by its restriction to the maximal torus T . A representation
V contains, as a dense subspace, the sum (v,n) rot V (v,n) of
subspaces on which this maximal torus acts by the indicated character.
This direct sum is recorded in terms of its character V(t,z,q)
t (v,n) V (v,n) q nz v. A priori this is merely a suggestive
way to record the map of sets align rot (v,n) V (v,n) . align
The requirements that V have bounded negative energy and that
V (k) is finite-dimensional for each k guarantee that V may be
viewed as an element of the ring t( ) (the notation refers to
the group ring on the character group etc.; the t records the
fixed character by which acts). enumerate Remark rem:finiteness
The restriction it:finite-V-k of Definition def:Representation
is also not standard. However, if representations are defined
without it, then the irreducible representations still satisfy
it:finite-V-k of def:Representation , and every representation
is essentially equivalent to a sum of irreducible representations.
The reason for the restriction is to make it easy to define the
tensor product of representations; see the discussion of Adams
operations in section sec:isogenies-and-reps . Remark Definition
def:RepkLG Denote by (a,E) (LG) the Grothendieck group of equivalence
classes of representations of LG of level E and type a . Definition
Relationship to theta functions sec:relationship-to-theta The
group T is a maximal torus of E a for any a ; we write T a if
we wish to distinguish the group in which it sits. Let (a) N(T
a ) T a be its Weyl group; it is called the affine Weyl group
. It acts on the maximal torus, and the character V is invariant.
The link between the representation theory of LG and the space
G comes from this invariance. Before studying (a) and its action
on the maximal torus, observe that we can identify ,T with a
subgroup of LG . Lemma t-le:Waff-shape Let W be the Weyl group
of T in G . Then the Weyl group (a) is isomorphic to W . The
element n acts on Ta by the formula n(z,q) (q an z,q). Lemma
proof This is standard. For example, Proposition (5.1.2) in PressleySegal:Loo
shows that is the Weyl group of T in LG . It is easy to see that
passing to LG a has no effect on the argument. As E is a central
extension, the action lifts to E , and the Weyl group can be
no larger. The formula for the action of on Ta is align n(z,q)(-n)
(n,1) (z,q) (n,-1) ((q a) nz( -n ),q) (q an z,q) (q an z,q).
align proof Because of Lemma t-le:Waff-shape , we drop the a
and revert to the usual notation for the affine Weyl group. Definition
def:degree-2 If A and B are abelian groups then a map : A B of
degree 2 is a map of sets such that (x y z) - (x y)- (x z) -
(y z) (x) (y) (z) - (0) 0. It is pointed if in addition (0) 0
. Definition Proposition prop:Waff There are a W -invariant pointed
map of degree 2 (a,E) : and a W -invariant symmetric bilinear
map I (a,E) : , related by the formula equation eq:phi-I-relation
(u v) (u) (v) a I (u,v), equation such that the action of n on
T a is given by the formula equation e:CocharsAction n (t,z,q)
( q (n) z I (n) t,q an z,q). equation In the formula, I (n) refers
to the element of , obtained from the adjoint of the bilinear
map I . Proposition proof Certainly there is some pair of pointed
functions align : : align such that n ( t,z, q) (q (n) z (n)
t, q an z,q). The associativity and commutativity of the action
first give that is a homomorphism, and so the adjoint of a bilinear
map I , and second that and I are related by eq:phi-I-relation
. The W -invariance can be seen by examining the equation (wn)(
t,wz, q) w(n( t,z, q)). proof Corollary t-co:char-is-theta If
V is a representation of level E and type a , then equation eq:transf-of-chars
V ( t,q an z, q) q -(n) z -I (n) V ( t,z, q). equation Corollary
proof Since V is a character we have V (n( t,z, q)) V ( t,z,
q). The left-hand side is V (q (n) z I (n) t,q an z, q). Since
acts by its standard character, this is in turn is equal to q
(n) z I (n) V ( t,q an z, q). proof Equation eq:transf-of-chars
is typical of the transformation formulae of theta functions.
Namely, recall the notations align T , T . align Let F be a p
-adic field or , and suppose that q F is an element with q 1
. The exact sequence 1 q a (F) (q a) (F) 1 of Theorem t:TateCurveII
results in an exact sequence equation eq:EGcovering 1 q a T(F)
T(q a) (F) 1, equation where the group q a an q n is isomorphic
to the subgroup . Proposition prop:Waff says that the action
of n on T(F) is translated by q an . Define T(F) e (F) by the
formula equation e:cocycle e(n,z) q -(n) z -I(n) . equation Then
the formula n(t,z) (e (n,z)t, zq an ) defines an action of on
F T(F) covering the action on T(F) , and gives rise to a line
bundle equation eq:basic-line-bundle FF (t,z) (e (n,z)t,zq an
) equation over T(q a) . A section of this line bundle is a function
f on T(F) which satisfies equation e:transformationformula f(zq
an ) e(n,z) f(z). equation Definition def:L Let (a,I,) be a line
bundle over T (q a) given by eq:basic-line-bundle . Let (a,I,)
be the group of power series f ( ) which satisfy f(zq an ) q
-(n) z -I (n) f (z,q) for n (see the appendix). If E is a central
extension of a loop group G , then we write (a,E) for (a,I (a,E)
, (a,E) ) and (a,E) for (a,I (a,E) , (a,E) ) . Definition Suppose
that F is a non-archimedean field and q is a unit of F with q
1 . Let ((a,E)) (q) denote the global analytic sections of the
resulting line bundle over (q) . In the cases of interest, it
turns out that ((a,E)) (q) F (a,E); see Theorems thm:FinalForm
and C:ETRLT . In any case, the map which assigns to a representation
of LG of level E and type a its character, may be viewed as a
homomorphism of -modules (a,E) (LG) (a,E). Of course, the characters
are also invariant under the action of the Weyl group W of G
. This group acts on (a,E) via its action on , and we denote
by (a,E) W the subgroup of power series which are invariant under
this action. We summarize the proceeding discussion as Proposition
t-pr:char-is-theta The character may be viewed as a homomorphism
of -modules ( a,E) (LG) (a,E) W. Proposition The Kac character
formula shows that this map is often an isomorphism. We treat
the case of simple and simply connected groups in section sec:simple-simply-connected
and the case of a compact torus in section sec:LT . Simple and
simply connected groups sec:simple-simply-connected The universal
central extension When G is simple and simply connected, any
central extension E of LG is G -split, and is completely determined
by the bilinear form I (1,E) of Proposition prop:Waff . In fact,
there is a universal central extension LG such that any other
is canonically isomorphic to h LG for some h , and the central
extension LG is canonically G -split. The bilinear form I associated
to LG is the smallest W -invariant bilinear form I such that
I (n,n) 2 for n (see Looijenga:RootSystems , 4.4 PressleySegal:Loo
), and the map is just (n) 12I (n,n). Definition def:Rep-of-level-k
Suppose that G is simple and simply connected. A representation
of LG of level k is a representation of LG of level k LG and
type 1 . Definition It is natural in this case to consider a
representation of level k as a representation of LG on which
acts by , and to write the character as (t,z,q) t k v,n V (v,n)
z vq n. We use the abbreviations align k (LG) (1,k LG ) (LG)
, (G) (1, LG ) , (G) k (1,k LG ) . align For k the line bundle
(1,k LG ) is just (1,G) k , and (G) k is the group of functions
f ( )((q)) which satisfy f(zq n,q) q -k(n) z -kI(n) f(z,q) for
n . For v , let g v,k g v,k,I be the function g v,k (z,q) n q
k(n) n,v z kI(n) v . Then according to Lemma app:gInGammak ,
g v,k is an element of (G) k , and Proposition app:gvbasis becomes
Proposition prop:gvbasis If S is a set of coset representatives
for k , then the functions g s,k for sS are a basis for (G) k
over . If F is a non-archimedean field and qF with q 1 , then
((G) k) (q) F (G) k. Proposition Sections invariant under the
Weyl group Looijenga Recall that (G) k W is the submodule of
(G) k which is fixed under the action of the Weyl group W of
T in G . We similarly define (G) k -W to be the submodule of
W -anti-invariant elements sections, i.e., elements s such that
w s (w) s for w W . Proposition prop:gvbasis parametrizes a basis
for (G) k by a fundamental domain for the action of k on (via
the inclusion induced by I ). We obtain a basis for k W once
we have a fundamental domain for the action of W on k , or, equivalently,
a fundamental domain for the action of on . The fundamental alcove''
is a fundamental domain. The results described from this point
until the character formula were obtained in Looijenga:RootSystems
. The fundamental alcove and the highest weight Alcove Let R
be the roots of G , let R be a set of positive roots, let 1,,
r be a basis of for R consisting of positive roots, and let i
be the coroot corresponding to i . The subset Ch given by Ch
: v v, i 0, 1 i r . is a fundamental domain for the action of
W on . Let be the highest root. The fundamental alcove of level
k is the simplex (in Ch ) P k v v, i 0 and I(,v) k . Proposition
CMk The set P k is a fundamental domain for the action of W k
on . Proposition For P k define A ,k (z) : w W (w) g ,k (wz).
Theorem LooijengasBasis The functions A ,k form a --basis for
(G) k -W . Theorem In order to describe the important basis for
(G) k W , we need a little more notation. enumerate 1. Let be
the coroot corresponding to . 2. Define non-negative integers
g i by g 1 1 g r r . 3. Set equation eq:dual-cox-g g 1 g i. equation
4. Let i be the basis of dual to i . 5. Let i. 6. Let equation
eq:Phi (q) n1 (1-q n). equation enumerate In terms of these,
let A(z,q) z (q) r R (1 - z - ) n 1 R (1 - q nz ). Proposition
LooijengasThetaA The function A is an element of (G) g -W . It
freely generates (G) -W as a module over (G) W . Proposition
The Kac character formula CharacterFormula The set P k also parametrizes
the irreducible representations of LG of level k . For P k ,
the Kac character formula records the character ,k of the representation
corresponding to . KacCharacterFormula KacCharFormula The character
of the representation of level k corresponding to P k is ,k (
t,z, q) t k A ,k g A. KacCharacterFormula See, for example, chapter
13 of Kac:Book . Corollary t-co:Grojnowski If G is a simple and
simply connected group, then the character map induces an isomorphism
k(LG) (G) k W. Corollary proof Theorem KacCharFormula together
with Theorem LooijengasBasis and Proposition LooijengasThetaA
show that the characters ,k for P k are a -basis for the space
(G) k W of W -invariant elements of (G) k . proof Corollary t-co:Grojnowski
is a form of Theorem thm:C over . To connect the result to the
Tate curve, suppose that F a is complete non-archimedean field,
and q is a unit of F with q 1 . Let (G) (q) be the resulting
W -equivariant line bundle over T (q) , and let G(;(G) k) (q)
denote Grojnowski's equivariant elliptic cohomology with coefficients
in (G) k (q) . Corollary t-co:Grojnowski together with Proposition
appgvbasisEC gives Theorem thm:FinalForm If G is a simple and
simply connected Lie group, then the character map establishes
an isomorphism F k(LG) G(,(G) k) (q). Theorem Remark rem:L-G-generally
Because the formal group of the Tate curve is , its behavior
in homotopy theory is essentially that of K -theory. It is encouraging
that the line bundle (G) , or at least its divisor class, can
be given over C , where C is any elliptic curve. We give a brief
explanation of this fact, for which we are indebted to Mike Hopkins
and the referee. Remark Each root of G determines a homomorphism
C C C whose kernel is a divisor on C . Let be the W -invariant
divisor R . Looijenga Looijenga:RootSystems observes that the
divisor class of is the same as that of the line bundle (G) g
, where g is defined in eq:dual-cox-g . M. Hopkins and the referee
have pointed out that the divisor class of (G) is also well defined
over any elliptic curve. Namely, let (C) be the map d () (C)
(e). In other words, if (-e) is the line bundle over C associated
to the divisor (e) , then d () is the divisor associated to the
line bundle (C) (-e). Lemma t-le:d-is-degree-2 The map d is a
pointed map of degree 2 as in Definition def:degree-2 : we have
multline eq:d-lambda-1-2-3 d ( 1 2 3) d ( 1) d ( 2) d ( 3) -
d ( 1 2) - d ( 1 3) - d ( 2 3) - d (0) 0 multline and d (0) 0.
Lemma proof The second equation is obvious. The first follows
from the theorem of the Cube. If is any line bundle over an elliptic
curve C , let () be the line bundle over C 3 whose fiber at (x,y,z)E
is () (x,y,z) x y z x y z x y x z y z 0 . The theorem of the
cube implies that this line bundle is trivial. In particular,
we may take (-e) ; then the left-hand side of eq:d-lambda-1-2-3
is just ( 1, 2, 3) () . proof Now if G and H are abelian groups,
then the set of pointed maps of degree 2 GH is isomorphic to
(G), H , where (G) is the degree-two subgroup of the symmetric
algebra on G ; the universal map is align G (G) g gg align (see
8 Breen:FonctionsTheta ). By Lemma t-le:d-is-degree-2 , the map
(C) extends canonically over u to a map () (C). It is easy to
check the following. Lemma t-le:Id-gamma-2 The group () is isomorphic
to the group of symmetric bilinear maps J : . Lemma It follows
that the symmetric bilinear map I I ( 1, LG ) gives an element
of () , and so a divisor class d (I) in (C) , functorially in
the elliptic curve C . In the case that C is the elliptic curve
q for q () , Looijenga shows that the class of the line bundle
(G) g is d (J) where J is the bilinear map J 12 R . In fact,
his methods can be used to prove the following. Proposition t-pr:d-I-is-L
The class in (C) of (G) is d (I) . Proposition proof Looijenga
shows that, in the transcendental theory Mumford:AbelianVarieties
, the image of the Chern class map (C) c 1 H 2 (C;) is () ()H
2 (C;). Moreover, using the fact that c 1((-e)) generates H 2
(C;) , it is easy to check that under this identification we
have c 1 ( (C) (-e)) . The universal property of implies that
() (C) c 1 () is the identity. It is also easy to check that
c 1 ((G)) I , so c 1 ((G)) I c 1 (d (I)). The proof is completed
by the following lemma. proof Lemma t-le:phi-doesnt-matter Let
K be a non-archimedean field or ; let q(K) ; and let C be the
elliptic curve K q . Let I : be a W -invariant, non-degenerate,
symmetric bilinear map. If and are two W -invariant pointed maps
of degree 2 from to such that align (u v) (u) (v) I (u,v) , (u
v) (u) (v) I (u,v), align then the line bundles (1,I,) and (1,I,)
over C are W -equivariantly isomorphic. Lemma proof Let - . Then
is a W -invariant linear map . Let v be the element such that
I (u,v) (u) for all u ; it is easy to check that wv v for wW
. The map (K ) K (K ) K given by the formula f (z,t) (zq v,t)
induces the desired isomorphism (1,I,) (1,I,). proof Representations
of LT sec:LT The description of the representation theory of
LG summarized in the previous section depends implicitly on a
detailed understanding of the representation theory of LT . Although
T is neither simple nor simply connected, by the time the dust
settles the representation theory of LT is essentially the same.
The group LT possesses a T -split central extension equation
e:CentExtT 1 LT LT 1, equation called the basic'' central extension,
whose inner product I I (1, LT ) induces an isomorphism I . Indeed,
after identifying r , where r is the rank of T , one may take
align I (x,y) i 1 r x iy i, (x) 12 i 1 r x i (x i-1). align The
representation theory is simpler since now Proposition prop:WaffCircle
The affine Weyl group of LT is just . Proposition Definition
def:Rep-of-level-k-T A representation of LT of level k is a representation
of LT of level k LT and type 1 . Definition Once again we simplify
the notation. align k (LT) (1,k LT ) (LT) , (T) (1, LT ) , (T)
k (1,k LT ) . align For k , (T) k is the group of functions f
( )((q)) which satisfy f(zq n,q) q -k(n) z -kI(n) f(z,q) for
n . For k 1 and v , let g v,k be the function g v,k (z,q) n q
k(n) v,n z kI (n) v . Proposition app:gvbasis gives Proposition
prop:gvbasisT If S is a set of coset representatives for k ,
then the functions g v,k for vS are a basis for (T) k . Proposition
Recall eq:Phi that (q) n1 (1-q n). KacCharacterFormula CharacterFormulaCircle
Let T be a torus of rank r . The irreducible representations
of LT of level k are parametrized by k . The character of the
irreducible representation corresponding to v is v,k ( t,z, q)
t k(q) -r g v,k (z,q). KacCharacterFormula proof See sections
9.5 and 14.3 in PressleySegal:Loo . proof Theorem C:ETRLT The
character map induces an isomorphism k(LT) (T) k. If F is a non-archimedean
field and q is a unit of F with q 1 , then this in turn induces
an isomorphism F k (LT) T(,(T) k) (q). Theorem Remark r:Heisenberg
The representation theory of the Hei -sen -berg group provides
a direct connection between and L . In addition to section 9.5
in PressleySegal:Loo the reader may wish to consult Mumford:TataIII
. Remark Calculations for L The loop group L has only one irreducible
representation of level 1 , called the basic representation'',
corresponding to 0 . In addition to the formula (z,q) (q) -1
k q 12k(k-1) z k provided by Theorem CharacterFormulaCircle ,
its character has a well known product expansion which may be
obtained from the following construction of the representation.
Let H denote the Hilbert space L 2(S 1,) . It carries an action
of L , but it is not of positive energy. However, H is polarized
H H H - according to the action of , and the basic representation
is a completion of the total exterior power equation eq:BasicIsExterior
(H - ). equation Since the character of H H - is equation eq:CharacterOfH
H - (z,q) k0 q kz k 0 q kz -1 , equation the character of is
equation eq:BasicCharProductExp (z,q) (1 z) k 0 (1 q kz)(1 q
kz -1 ). equation The Tate curve has a point of order 2 , namely
the point -1 in K q ; its pre-image in K is the set of points
-q . The formula eq:BasicCharProductExp shows that (z,q) , considered
as a function on K , vanishes to first order at these points
-q . It follows that the line bundle () is the line bundle defined
by the divisor (-1) on , and indeed one has the equation (z,q)
(-z,q) with given by eq:sigma . This observation was one starting
point of our investigation. Power operations and loop groups
part:POLG Isogenies and their effect on theta functions sec:isogenies-and-reps
In section sec:relationship-to-theta , we showed that a G -split
central extension E of a loop group LG determines a W -equivariant
line bundle (n,E) over T (q n) , and that the character of a
representation of level E and type n is naturally a W -invariant
section of (n,E) . In section sec:HeckeTate , we studied two
basic families of isogenies of the Tate curve, namely alignat
2 g 1,b : (q) (r) over D (1,b), g a,1 : (q) (q a) over D (a,1)
. alignat The notation (q a) makes implicit use of the specialization
map align s a q q a align as (q a) s a . For b 1 it is convenient
also to consider the specialization align D (1,b) s b, q q b
r q, align under which g 1,b becomes an isogeny g 1,b, : (q b)
(q). If I : and : are as in section sec:relationship-to-theta
, satisfying (u v) (u) (v) I (u,v), then it is easy to check
the following. Lemma t-le:Effect-of-isog-on-L subequations eq:EC-trans
align g 1,b, (1,I,) (b,bI,(b)) , g a,1 (a,I,) (1,aI,) , s a (1,I,)
(a,I,a), g a,1 s a (1,I,) (1,aI,a) (1,I,) a. align subequations
If f is a section of the line bundle (1,I,) , viewed as a function
f (z,q) , then subequations eq:EC-trans-fn align g 1,b, s b,
f (z,q) f (z,q) , s a f (z,q) f (z,q a). align If f is a section
of (a,I,) , then equation g a,1 f (z,q) f (z a,q). equation If
f is a section of (1,I,) , then equation g a,1 s a f (z,q) f
(z a,q a). equation subequations Lemma Isogenies and representations
The isogenies of the Tate curve are related to a family of homomorphisms
f a,b : LG ab b a LG 1 generated by align a (,) (, a) , b (z)
(z b). align If E is a G -split central extension, then there
is a homomorphism Eab b Eab E1 covering f a,b . It is easy to
check the following. Lemma t-le:LG-f-a-b If I I ( 1,E) and (
1,E) , then align I (ab, b E) ab I , (ab, b E) () a(b). align
Moreover, if V is a representation of level E and type 1 , then
align (a,E) s a (1,E) , (b, b E) g 1,b s b,1 (1,E). align If
V is a representation of LG of level E and type c , then align
a V s a V, b V g 1,b V. align Lemma Comparing Lemma t-le:LG-f-a-b
with Lemma t-le:Effect-of-isog-on-L , we have Proposition t-pr:isog-and-reps-I
enumerate () The pull-back of characters along the magnification
of loops b corresponds to the pull-back of sections of the line
bundle (1,E) along the isogeny g 1,b,1 . The pull-back of characters
along the characters along the homomorphism a corresponds to
the specialization of sections of the line bundle (1,E) under
the map s a (q) q a . enumerate Proposition There are two families
of isogenies missing from Proposition t-pr:isog-and-reps-I :
the isogeny (q) g a,1 (q a) and the isogenies (q b) g 1,b, (q).
We deal with g 1,b, in the next section, and turn to g a,1 .
The Adams operation From the formula g a,1 s a f (z,q) f (z a,q
a) of Lemma t-le:Effect-of-isog-on-L , it is clear that the isogeny
g a,1 should correspond to the Adams operation. We simply observe
that such an operation can be defined. If V and W are two representations
of LG of type 1 and level E and F respectively, then there is
a tensor product of V and W : by assumption the space V W (k)
i j k V (i)W (j) is finite dimensional, and zero for k sufficiently
small. There is a natural inner product on equation eq:V-otimes-W-finen
k V W (k), equation and we may define V W to be the Hilbert space
completion of eq:V-otimes-W-finen with respect to this inner
product. It is a representation of LG of level EF . In particular,
if V is a representation of level E , then V a is a representation
of level a E ; in fact it is a representation of aa E . As such
it may be decomposed as a representation of a ; the usual procedure
of evaluating on the class of an a -cycle yields a map (1,E)
(LG) a (1,a E) . The usual argument shows that the effect on
characters is given by the formula aV (z,q) V (z a,q a) g a,1
s a V. Proposition t-pr:isog-and-reps-II-adams The Adams operation
a on characters of representations corresponds to the operation
a,1 of section sec:HeckeTate ; namely, aV g a,1 s a V. Proposition
Finite subgroups of the rotation group magnification of loops
sec:Trotn Representations of type b sec:type-b A key point about
representations of type b is Lemma t-le:crot-b The subgroup of
b of b -order torsion is contained in the center of E b . Lemma
Thus a representation of level E and type b may be decomposed
as a representation of b . After a choice of map align b b q
b , align we may write the character of V as a representation
of b E as V (z,q,) V (z,q) k- q k kp k, with p k . Magnification
of loops sec:mag In particular, if V is a representation of LG
of level E and type 1 , then b V is a representation of level
b E and type b . Choose a map b b ; let us name the map by the
image of equation eq:q-to-zeta q b . equation Then the character
of f 1,b V as a representation of b b E is f 1,b V (z,q,) V (z,q);
it is a section of (b,bI,(b)) . Proposition t-pr:isog-and-reps-III-mag
The process of pulling back a representation along f 1,b and
then viewing it as a character of b b E using eq:q-to-zeta corresponds
to the operation 1,b, of section sec:HeckeTate ; namely, f 1,b
V (z,q,) g 1,b, s b, V. Proposition Remark rem:b-star-and-b-star
According to Lemma t-le:LG-f-a-b , I ( b, b E) bI . In the case
that G is simple and simply connected, it follows that the central
extension b LG is canonically isomorphic to b LG , so b V is
a representation of LG of level b '' in the sense of Definition
def:Rep-of-level-k . In fact, this is also the case when LG is
the basic'' central extension of a simply-laced group 9 PressleySegal:Loo
. Remark Decomposition of representations of type b If E b (V)
is a representation of type b , let j :E U (V j) be the the summand
on which b acts by the character q j b . Proposition t-pr:V-j
The representation q -j j factors through E b b E 1. Proposition
Let (j) or simply V (j) denote the resulting representation.
If V k- q k p k with p k , then V (j) k- q k p bk j . Theta functions
app:thetafunctions The theta functions in section sec:simple-simply-connected
are equivalent to those in Kac:Book and Looijenga:RootSystems
, but our notation is different from theirs. To aid the reader
with the translation, we give a proof (along the lines of Looijenga:RootSystems
) of Proposition prop:gvbasis , which applies as well to Proposition
prop:gvbasisT . Notation app:notation align L : a free abelian
group of rank r, : a non-negative map of degree 2 L , I : a non-degenerate
symmetric bilinear form L L , such that (u v) (u) (v) I(u,v)
, : L , : L , q L : L q nq k n L; k , M : L, , ,: the evaluation
LM . align The situation of the rest of the paper can be recovered
by taking L . In section sec:simple-simply-connected we have
() 12I (,). In the situation of section sec:LT , we have () 12
i 1 r i ( i-1) in terms of an isomorphism r . 1 As usual, we
denote by z i n i the element of which might also be written
n iz i. The adjoint of I is an injection equation app:Iadjoint
xy L ( - r I M xy equation which we also denote by I . For example,
if n L , then we obtain a homomorphism L I(n) ; for z z i n i
we have align z I(n) I(n)(z) I(n)(z i n i ) z i I(n,n i) . align
The behavior of and I is quadratic in the following sense. Lemma
t-le:I-phi-quadratic Suppose that k is an integer greater than
0 . For each N and vM , the set L ,v k() - kI (,) N is finite.
Lemma proof This is standard. Let e 1,,e r be a basis for L ,
and define rational constants a i , b ij , c i , and v i by the
formulae align a i 2I (e i,e i) , b ij kI (e i,e j) , c i k(e
i) , v i e i, v . align Since I is positive definite, the a i
are strictly positive. If ie i , then it is easy to check that
,v k() - kI (,) i 1 r (v i - a i c i) i - 1i jr b ij i j - ia
i i 2. The set of for which this quadratic polynomial is positive
is finite. proof A map of sets c : M will be recorded as a formal
sum f(z,q) n v M c(n,v)q nz v. Definition def:app-gamma We denote
by (k,I,) or just the k group of functions c : M which satisfy
the following: enumerate () There is a b such that c (n,v) 0
for all vM and n b . The formal sum f corresponding to c satisfies
the equation equation app:fmltransformationformula f(zq ) q -k()
z -kI() f(z) equation for all L . enumerate Definition Lemma
app:thetafncoeffs The equation app:fmltransformationformula is
equivalent to c(n,v-kI ()) c(n ,v k() - kI(,),v) for n and vM
. Lemma proof The expansion of the left-hand side of f(zq ,q)
q -k() z -kI() f(z) is f(zq ,q) n v M c(n,v) q n ,v z v. The
right-hand side is q -k() z -kI(w) f(z) n v M c(n,v)q n-k() z
v-kI() . Then, comparing coefficients, we have align c(n,v-kI())
coefficient of q n , v-kI () z v-kI () on left coefficient of
q n , v -kI (,) z v-kI () on left c( v, n ,v k() - kI(,)) on
right. align proof Proposition app:Gamma0 0 . Proposition proof
First, for each v we know that c(n,v) 0 for n sufficiently small.
When k 0 , Lemma app:thetafncoeffs says that c(n,v) c(n ,v ,v
) for all L . Unless v 0 , it follows that c(n,v) 0 . proof Proposition
app:AutoPoly For k0 , if f k , then f is an element of the ring
( M ). Proposition proof Write f(z,q) n,v c(n,v) z vq n. Let
S be a set of coset representatives for M kI(L) . It suffices
to show that for each n and s S , the set L c(n,s- kI()) 0 is
finite. Lemma app:thetafncoeffs shows that c(n,s-kI(w)) c(n ,s
k() - kI(,),s). Lemma t-le:I-phi-quadratic and the assumption
that f k shows that the coefficient on the right is non-zero
for only finitely many w . proof A choice S of coset representatives
for M kI(L) determines a basis for k . For v M , let g v,k be
the function g v,k (z,q) g v,k (z,q,I) nL q k(n) n,v z kI (n)
v . Lemma app:gInGammak g v,k is an element of k . Lemma proof
For L we have align g v,k (zq ,q) nL q k(n) n,v z kI (n) v q
kI (n,) ,v . q -k() nL q k(n ) n ,v z kI (n) v q -k() z -I ()
nL q k(n ) n ,v z kI (n ) v q -k() z -I () g v,k (z,q). align
proof Proposition app:gvbasis If k 0 and if S is a set of coset
representatives for M kL , then the functions g s for sS are
a basis for k over . Proposition proof The g s,k for s S are
certainly linearly independent over ; just compare coefficients
of q k(0) z s for sS . Now suppose that f k, and write f(z,q)
n,v c(n,v) z vq n. Define a s by the formula a s n c(n,s)q n.
Then sS a sg s,k and f both live in k . The coefficient of z
sq n in their difference is 0 . By Lemma app:thetafncoeffs ,
they are equal. proof Now let F be a complete, non-archimedean
field, and let qF be an element with q 1 : we obtain a line bundle
(I,) over the variety L(q) . Proposition appgvbasisEC If k 0
and if S is a set of coset representatives for M kI (L) , then
the functions g s,k for sS are an F -basis for the global sections
of (I,) . Proposition proof In view of Proposition app:gvbasis
, what remains is to remark that the power series g v,k (z,q)
converges since the q 1 and the coefficient of q N is a Laurent
polynomial in r variables of degree on the order of in each variable.
proof Calculations for L sec:CalcLT ]]></doctext>
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