]]> Let be a non-archimedean local field of residual characteristic . Then has tamely ramified self-contragredient supercuspidal representations if and only if or is even. When such representations exist, they do so in abundance. bushnell-kutzko:gln-book C. J. Bushnell and P. Kutzko, The admissible dual of GL N via restriction to compact open subgroups , Ann. of Math. Studies, vol. 129, Princeton University Press, 1993. 94h:22007 gelfand-graev I. M. Gelfand and M. I. Graev, The group of matrices of second order with coefficients in a locally compact field and special functions on locally compact fields , Uspekhi Mat. Nauk 18 (1963), 29--99. 27:5864 gelfand-kazhdan I. M. Gelfand and D. A. Kazhdan, Representations of GL (n,K) where K is a local field , Lie groups and their representations: Proc. Summer School on representation theory, Hungary, 1971 (I. M. Gelfand, ed.), 1975, pp. 95--118. 53:8334 howe:gln R. Howe, Tamely ramified supercuspidal representations of GL n , Pac. J. Math. 73 (1977), no. 2, 437--460. 58:11241 lusztig:chars-finite G. Lusztig, Characters of reductive groups over a finite field , Ann. of Math. Studies, vol. 107, Princeton University Press, 1984. 86j:20038 morris:level-zero L. Morris, Level zero G -types , preprint. moy:thesis A. Moy, Local constants and the tame Langlands correspondence , Amer. J. Math. 108 (1986), 863--930. 88b:11081 moy-prasad:K-types A. Moy and G. Prasad, Unrefined minimal K -types for p -adic groups , Inv. Math. 116 (1994), 393--408. 95f:22023 moy-prasad:jacquet , Jacquet functors and unrefined minimal K -types , Comm. Math. Helv., to appear. 96:07 sally:bessel P. J. Sally, Jr., Invariant subspaces and Fourier-Bessel transforms on the -adic plane , Math. Ann. 174 (1967), 247--264. 58:28317 serre:local-fields J.-P. Serre, Corps locaux , Hermann, 1962. 27:133 shahidi:twisted F. Shahidi, Twisted endoscopy and reducibility of induced representations for p -adic groups , Duke Math. J., 66 (1992), 1--41. 93b:22034

C. J. Bushnell and P. Kutzko, The admissible dual of via restriction to compact open subgroups, Ann. of Math. Studies, vol. 129, Princeton University Press, 1993. MR 94h:22007

I. M. Gelfand and D. A. Kazhdan, Representations of where is a local field, Lie groups and their representations: Proc. Summer School on representation theory, Hungary, 1971 (I. M. Gelfand, ed.), 1975, pp. 95-118. MR 53:8334

R. Howe, Tamely ramified supercuspidal representations of , Pac. J. Math. 73 (1977), no. 2, 437-460. MR 58:11241

L. Morris, Level zero -types, preprint.

A. Moy and G. Prasad, Unrefined minimal -types for -adic groups, Inv. Math. 116 (1994), 393-408. MR 95f:22023

-, Jacquet functors and unrefined minimal -types, Comm. Math. Helv., to appear. CMP 96:07

P. J. Sally, Jr., Invariant subspaces and Fourier-Bessel transforms on the -adic plane, Math. Ann. 174 (1967), 247-264. MR 58:28317

F. Shahidi, Twisted endoscopy and reducibility of induced representations for -adic groups, Duke Math. J., 66 (1992), 1-41. MR 93b:22034
]]> 1997 American Mathematical Society Proceedings of the American Mathematical Society Proc. Amer. Math. Soc. proc 125 08 19970801 December 1, 1995 February 12, 1996 2471 2471-2479 9 S0002-9939-97-03786-6 10.1090/S0002-9939-97-03786-6 0002-9939 1088-6826 English 22E50 20G05 11F70 1991 http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9939-97-03786-6 Suppose G is a symplectic or split special orthogonal group over a p -adic field F (of either zero or positive characteristic), and P is a maximal parabolic subgroup of G with Levi factor M isomorphic to n(F) . If is a unitary supercuspidal representation of M , then we can form the induced representation Ind P G of G . In order for to be reducible, it is a necessary condition that be self-contragredient. (See shahidi:twisted , which also investigates sufficiency conditions when F has characteristic zero.) For if is reducible, this implies that the nontrivial element w of the Weyl group of M in G fixes the isomorphism class of . However, w acts on M by tm , and so by a result of Gelfand and Kazhdan Theorem 2 gelfand-kazhdan , is self-contragredient. Thus, it is of interest to know which p -adic general linear groups have self-contragredient supercuspidal representations, and to have examples. We exploit a construction of supercuspidal representations due to Howe howe:gln , and call the resulting representations tamely ramified . (Moy moy:thesis has shown that if p and n are relatively prime and F has characteristic zero, then all supercuspidals of n(F) are tamely ramified.) Since these representations are constructed from admissible'' characters of extension fields, it is necessary to examine the structure of these fields and their characters in some detail. In sec:polar , we write down the p -adic analogue of polar coordinates. In the two-dimensional, ramified case, this is due to Gelfand-Graev gelfand-graev and Sally sally:bessel . The p -adic analogue of the unit circle comes equipped with a natural filtration, which we study in sec:filtration . Since an extension field E of F can have many intermediate subfields, E can have many polar decompositions, all of which we need to consider. We compare them in sec:family . In sec:definition , we give necessary and sufficient conditions for a character to parametrize a self-contragedient supercuspidal representation. This allows us to prove our main result, Theorem thm:existence , which implies that n(F) has tamely ramified self-contragredient supercuspidal representations if and only if n or p is even, in which case one can attach such representations to most tamely ramified extensions E F of degree n . For the sake of explicitness, we count the examples of depth zero in sec:depth-zero . This leaves open the question of whether self-contragredient supercuspidal representations (not tamely ramified) exist when p and n are odd, and p n . In order to find the answer, one has to deal with wildly ramified extension fields of F . But in this situation, the supercuspidal representations of n(F) are no longer parametrized by admissible characters. Therefore, although much of our study of admissible characters carries over (in a more complicated form) to the wild case, we have restricted ourselves to the tame case wherever convenient. I wish to thank Paul Sally and Mark Reeder for very helpful discussions, and David Goldberg for clarifying for me the connection between self-contragredience and reducibility. Notation and conventions For any p -adic field F , let F denote its normalized valuation, k F its residue field, and q F the order of k F . We denote the prime ideal in F by F , and a uniformizing element by F . We let U F be the group of units, and let U F,i 1 F i for all i . We can (and will) identify the multiplicative group k F with the group of roots of unity in F of order prime to p . By a character of F , we mean a continuous homomorphism F . For any finite-dimensional field extension E F , let C E F denote the kernel of the norm map E F from E to F . For i 0 , let C E F,i C E F U E,i . Let C E F,0 C E F . We will sometimes write C E F (0) for C E F,1 . As usual, we let e(E F) denote the ramification degree of E F . The extension E F is tamely ramified (or tame ) if e(E F) is relatively prime to p . There is a canonical decomposition equation eqn:jordan C E F C E F C E F (0) , equation where equation eqn:barC C E F xk E k E k F (x e(E F) ) 1 . equation If A and B are elements or subsets of a group, then A,B denotes the subgroup generated by A and B . Polar decomposition sec:polar Let E F be a finite extension of p -adic fields. Then E is almost a direct product of F and C E F . The purpose of this section is to make this statement more precise in two special cases. lemma lem:polar Let E F be an extension of degree n . Then the norm map E F induces an isomorphism U E,1 C E F (0) U F,1 E F (U E,1 ) (U F,1 ) n . lemma proof Let xU E,1 , and suppose E F (x)(U F,1 ) n . Pick yU F,1 such that y n E F (x) . Then xyC E F (0) , so xC E F (0) U F,1 . The surjectivity of the map is clear. proof Note that if E F is tame, then E F (U E,1 ) U F,1 , as we will see from Lemma lem:norm-tame . If, furthermore, (n,p) 1 , then (U F,1 ) n U F,1 and C E F (0) U F,1 1 , so we have a direct product decomposition U E,1 C E F (0) U F,1 . prop prop:polar-unram Let E F be an unramified extension of degree n . Then E F induces an isomorphism E C E F (0) k EF E F (E) k F(F) n U F,1 (U F,1 ) n . prop proof Similar to that of the lemma. proof In particular, if (n,p) 1 , we get a direct product decomposition E C E F (0) k EF. prop prop:polar-tot-tame Let E F be a totally and tamely ramified extension of degree n . Write E F( E) , where E n is a prime in F . Then E C E F (0) F, E. prop proof Similar to that of the lemma. proof The filtration C E F,i i 0 sec:filtration We want to find the successive quotients of this filtration. We start by recalling how E F behaves with respect to the filtration U E,i . lemma lem:norm-tame Let E F be a tamely ramified extension of ramification degree e . Then E F (U E,i ) U F, (i-1) e 1 . lemma proof If E F is unramified, then this is just V, Prop. 3 serre:local-fields . If E F is Galois and totally ramified, then this is a special case of V, 6, Cor. 3 serre:local-fields . Therefore, the proposition is true if the ramified part of E F is Galois. By adjoining roots of unity to E , we can obtain a field E' such that E' E is unramified and E' F (and thus its ramified part) is Galois. Then, from the previous paragraph, E' E (U E',i ) U E,i and E' F (U E',i ) U F, (i-1) e 1 , so align E F (U E,i ) E F E' E (U E',i ) E' F (U E',i ) U F, (i-1) e 1 . align proof prop prop:quotient-size For any extension E F and any i 0 , C E F,i C E F,i 1 q E E F (U E,i ) E F (U E,i 1 ) . For i 0 , replace q E by q E-1 . prop proof For any i0 , we have a commutative diagram U E,i 1 C E F,i 1 ((- r -.4ex d U E,i C E F,i d E F (U E,i 1 ) ((- r -.4ex E F (U E,i ) where the horizontal arrows are induced by inclusion, and the vertical arrows are isomorphisms induced by the norm map. Thus, E F (U E,i ) E F (U E,i 1 ) U E,i U E,i 1 C E F,i C E F,i 1 , which implies our conclusion. proof cor cor:filtration-quotients-tame If E F is tame and i 0 , then C E F,i C E F,i 1 cases q E q F if e(E F) i , q E otherwise . cases Also, C E F,0 C E F,1 q E-1 q F-1 (e,q F-1). cor proof The first statement is immediate from Lemma lem:norm-tame and Proposition prop:quotient-size , and the second follows from the latter and eqn:barC . proof The family of subgroups C E L,i ELF sec:family lemma lem:add-trace If E LL' is separable, then E L E L' E LL' . lemma This actually makes sense for any field E ( p -adic or not) and any subfields L and L' such that E (LL') is finite-dimensional and separable. proof (x,y) E LL' (xy) is a nondegenerate, symmetric, LL' -bilinear form on E . For any subset SE , let S xE E LL' (xs) 0 for all sS . Then for any intermediate field EKLL' , align E K E K (x) 0 K E K (x) 0 K E K (x) K LL' (since K E K (x) is a K -subspace of K ) E LL' (xK) 0 K . align Therefore, we need to show L L (LL') , which is elementary. proof prop prop:prod-leveli Suppose that E LL' is tame, and i 0 . Then C E L,i C E L',i C E LL',i . prop proof It is clear that the left-hand side is contained in the right. Let c c 0C E LL',i . Then c 1 x 1 , with x 1 E i , and E LL' (c) 1 . But E LL' (c)1 E LL' (x 1) U E,2i , so E LL' (x 1)0 E 2i . From the lemma and lem:norm-tame , we can find t 1 1 y 1 and t' 1 1 y' 1 in C E L and C E L' , respectively, so that y 1 y' 1x 1 E 2i . Therefore, t 1t 1'c 0 C E LL',2i . Let c 1 c 0t 1t 1 -1 C E LL',2i . Repeating this process, we may write c j 1 t j j 1 t' j, where the infinite products converge, and they lie in C E L,i and C E L',i respectively. proof Definition of admissible and self-contragredient characters sec:definition Let E F be a finite extension of p -adic fields. For any character of E , define the level of to be the smallest nonnegative integer i such that U E,i is trivial. defn def:admissible A character of E is admissible over F if enumerate (1) C E L is nontrivial for all ELF , and C E L (0) is nontrivial for all ELF such that E L is ramified. enumerate defn Note that in the definition we may restrict ourselves to maximal subfields L . If E F is a tamely ramified extension of degree n and is an admissible character of E , then let denote the supercuspidal representation of n(F) that arises from via the Howe construction. (For details, see howe:gln or moy:thesis . While the latter uses a blanket assumption that (n,p) 1 , the section devoted to the construction of supercuspidal representations works for any tamely ramified E .) From moy:thesis , we know that all supercuspidal representations of n(F) arise in this way if (n,p) 1 and F has characteristic zero. Two characters and ' of E and E (respectively) are conjugate if there is an F -isomorphism E E' such that ' (sometimes denoted ' ). prop Howe prop:conj If and ' are admissible characters of E and E , respectively, then ' if and only if and ' are conjugate. prop prop prop:conditions Let E F be tame, and let be an admissible character of E . Then is self-contragredient if and only if one of the following conditions holds: enumerate (a) there is some ELF such that E:L 2 and E L (E) is trivial; (b) p 2 and has order two. enumerate prop proof Note that . In the case where has level one, this follows from the analogous fact for Deligne-Lusztig virtual representations lusztig:chars-finite . Otherwise, it follows from the fact that is (unitarily) induced from an extension of to a subgroup of E:F (F) containing an embedded image of E . (See howe:gln for details.) From Proposition prop:conj , if and only if for some F(E) . In particular, 2 . Suppose is nontrivial and has odd order. Then 2 implies that , so is trivial on (x) x xE , which equals C E E by Hilbert's Theorem 90. But this contradicts the admissibility of . Suppose that is nontrivial and has even order. Then 2 so, by reasoning similar to that used above, is trivial on C E E 2 . The admissibility of then implies that 2 1 . Let L E . Then E:L 2 , and for all xE , ( E L (x)) ((x)x) (x)(x) 1. This reasoning also works in reverse. Now suppose 1 . Then has order two. The fact that p must equal 2 in this case follows from the next result. proof prop Let E F be any extension of p -adic fields, with p odd. Then E has no real-valued admissible characters over F . prop proof Suppose that E F is ramified and is a real admissible character of E . Then is nontrivial on C E F (0) . But this is a pro- p -group, so the order of any of its characters must be a power of p , a contradiction. Now suppose that E F is unramified. Then it will be enough to show that C E F (E) 2 , since this will imply that all real characters of E are trivial on C E F . Write E F() , where is a root of unity. Choose a minimal positive r such that r is congruent mod to an element of C E F , i.e., such that the corresponding element r in k E lies in C k E k F . It is enough to show that r is even. In fact, r k E C k E k F k F , which is even. proof defn By an abuse of language, let us call a character self-contra -gredi -ent if it satisfies either of the conditions of prop:conditions . defn Thus, we are interested in the existence of self-contragredient admissible characters. These characters are necessarily unitary, as are the corresponding supercuspidal representations. cor cor:conditions Let E F be a tamely ramified extension, and let L 1,L 2,,L r be the maximal intermediate fields. Then a character of E is self-contra -gredient and admissible if and only if both of the following conditions hold: enumerate (1) is nontrivial on C E L i for all i , and (2) is nontrivial on C (0) E L i for all i such that E L i is ramified, enumerate and one of the following conditions holds: enumerate (3) there is some 1ir such that E L i is quadratic and is trivial on E L i (E) , (3') p 2 and has order 2 . enumerate cor Existence sec:existence Here is a our main theorem. thm thm:existence Suppose that E F is any tame extension of degree n . Then E has self-contragredient admissible characters if and only if either p 2 or E is quadratic over some intermediate field. thm First we need a lemma. lemma lem:existence Let A be a topological abelian group, let C 1,,C r be closed subgroups, and let N be a closed subgroup that does not contain any C i . Then there exists a character of A such that is trivial on N and is nontrivial on every C i . lemma proof Replacing A by A N and each C i by its image in A N , we may reduce to the case where N is trivial. It is now enough to find a partition of 1,,r into subsets S j such that the groups H j iS j C i are all nontrivial, but have trivial pairwise intersections. For then we may choose any nontrivial characters j of each H j , let be the corresponding character of H 1 H 2A , and extend to A . To construct such a partition, let S 1 be any maximal subset of 1,,r such that iS 1 C i is nontrivial, and then proceed by induction. proof proof Proof of Theorem thm:existence If n and p are both odd, then it is clear from Corollary cor:conditions that for no tame extension E F of degree n does E have a self-contragredient admissible character. Suppose that L 0 is an intermediate field such that E L 0 is quadratic. From lem:existence , it is enough to show that E L 0 (E) does not contain any C E L i (0) , where L i is an intermediate field. Suppose on the contrary that C E L i (0) E L 0 (E) for some L i . Then C E L i (0) U L 0,1 . Since C E L 0 (0) U L 0,1 is finite, so is C E L 0 (0) C E L i (0) . This implies that, as a manifold over F , the product C E L i (0) C E L 0 (0) has dimension ( n - L i) (n-L 0) n. But from prop:prod-leveli , this product is C E L 0L i (0) , which has dimension at most n-1 , a contradiction. Now suppose that p is even. From lem:existence , it is enough to show that (E) 2 does not contain any C E L i (0) . Suppose that C E L i (0) (E) 2 for some L i . It is elementary that U E,1 (E) 2U E,2 , so C E L i,1 C E L i,2 . But this contradicts cor:filtration-quotients-tame . proof Examples at depth zero sec:depth-zero The depth of an irreducible representation is defined in moy-prasad:K-types . For our purposes, it will be enough to say that a representation has depth zero if it has nontrivial P -fixed vectors, where P is the maximal normal pro- p -subgroup of some parahoric subgroup P of n(F) . From work of Bushnell-Kutzko bushnell-kutzko:gln-book , Morris morris:level-zero , or Moy-Prasad moy-prasad:jacquet , all supercuspidal representations of n(F) of depth zero are tamely ramified in the sense we are using here. That is, they all arise via the Howe construction from admissible characters of level one. (Note that there is no restriction on n or p .) thm thm:depth-zero The number of self-contragredient supercuspidal representations of n(F) of depth zero is cases 0 if n is odd , 2n (q n 2 -1) if n is a power of 2 and p is odd , 2n (q n 2 ) if n is a power of 2 and p 2 , 2n S 1,,t (-1) S q F n (2 iS p i) otherwise, cases where p i 1i t is the set of odd prime divisors of n . thm proof We start by counting the self-contragredient admissible characters of level one of tame extensions E F of degree n . From Definition def:admissible , such characters can only exist when E F is unramified. No such characters can have order 2, so they can only exist when n is even, and they must satisfy condition (3) of Corollary cor:conditions . Let p 0 2 . For each 0i t , let k i be the intermediate field in k E k F such that k E:k i p i . Then the k i are the maximal intermediate fields. Let C i be the image in k E k 2 of C k E k i . Let q q F . Recall that for any positive integers b , c , and d , q b-1 divides q d(q bc -1) q bc d -q d . Therefore, equation eqn:gcd (q b-1,q bc d 1) (q b-1,q d 1). equation We have align C i C k E k i C k E k i k 2 (q n-1) (q n p i -1) ( q n-1 q n p i -1 ,q n 2 -1) (q n-1) (q n p i -1) ( q n-1 q n p i -1 , q n-1 q n 2 1 ) (q n p i -1,q n 2 1) q n p i -1 q n 2 1 (q n p i -1,q n 2 1) . align Using eqn:gcd , we can simplify this to q n 2 1 r i , where r i cases q n 2p i 1 if i 0 , 2 if i 0 and p is odd , 1 if i 0 and p 2 . cases For any subset S 0,,t , the product of all C i with iS has order q n 2 1 r i iS . Therefore, the number of characters of k E k 2 that are nontrivial on every C i is (q n 2 1) S 0,,t (-1) S r i iS . If n is a power of two, then this simplifies to q n 2 -1 if p is odd, and q n 2 if p 2 . Suppose n is even, but not a power of two. The terms involving subsets S containing 0 are all (-1) S r 0 . These terms cancel each other out, and we are left with (q n 2 1) S 1,,t (-1) S q n 2p i 1 iS , which simplifies to S 1,,t (-1) S q n (2 iS p i) . All self-contragredient admissible characters of level one of E over F arise by inflating such characters of k E to U E and extending to E . There are always two such extensions, as E can be sent to 1 . Two such characters give the same representation of n(F) if and only if they are in the same orbit of the action of (E F) on the characters of E . Our result follows from the fact that admissible characters lie in orbits of size n , from reasoning used in prop:conditions . proof To get concrete examples, take E F unramified of degree n , take any m that divides q n 2 1 but that does not divide any q n p i -1 (for example, m q n 2 1 ), take any character of k E (k E) m with trivial kernel, inflate to a character of U E , and extend to E by sending to either 1 or -1 . Given a self-contragredient admissible character of level one of E , it is easy to obtain examples at any higher level. As before, let L 0 be the intermediate field such that E L 0 is quadratic. Let be any character of C E L 0 (0) of level 1 . (If p 2 , we require that (-1) 1 .) Then extends to a character of U E,1 trivial on U L 0,1 . Extend this character trivially to U E , and then one can further extend to get a character on E that is trivial on E L 0 (E) . Then is a self-contragredient admissible character of level . to3em ,