# Unipotent representations and microlocalization

By Lucas Mason-Brown

## Abstract

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev [Duke Math. J. 159 (2011), pp. 99–143]. We explore the applications of this theory to unipotent representations of real reductive groups. For a unipotent representation of a complex group, we deduce a formula for the restriction to a maximal compact subgroup, proving an old conjecture of Vogan [Associated varieities and unipotent representations, Birkhäuser Boston, Boston, MA, 1991] in a large family of cases.

## 1. Introduction

Let be the real points of a connected reductive algebraic group. In Reference 1, Adams, Barbasch, and Vogan, following ideas of Arthur Reference 2,Reference 3, introduced a finite set of irreducible representations of , called special unipotent representations. We will recall their definition in Section 2.1. These representations are conjectured to possess an array of distinguishing properties (see Reference 1, Chp 1). For example:

They are conjectured to be unitary.

They are conjectured to appear in spaces of automorphic forms.

They are conjectured to generate (through various types of induction) all irreducible unitary representations of of integral infinitesimal character

Now let be a maximal compact subgroup. Any (nice) irreducible representation of decomposes as a -representation into irreducible components, each with finite multiplicity. The general philosophy of unipotent representations suggests that if is special unipotent, then these multiplicities should be ‘small’. In Reference 18, Conj 12.1, Vogan offers a conjectural description of the restriction to of a special unipotent representation (under some additional conditions). A little more precisely, he conjectures

Of course, the actual conjecture in Reference 18 is much more precise and involves a certain ‘codimension condition’ on the associated variety of , see Conjecture 2.7. In Corollary 5.2, we will prove Vogan’s conjecture in a large family of cases.

The main ingredient in our proof is a functor which ‘microlocalizes’ Harish-Chandra modules over a nilpotent -orbit . The construction of follows Reference 13, Sec 4, where a similar functor is constructed for Harish-Chandra bimodules.

Here is an outline of the structure of this paper. In Section 2, we will review some preliminary facts about primitive ideals, associated varieties, unipotent representations, and the localization of categories. We will also give a precise formulation of Vogan’s conjecture, cf. Conjecture 2.7. In Section 3, we will develop our main technical tool for proving Conjecture 2.7, namely a microlocalization functor for Harish-Chandra modules. In Section 4, we will relate Vogan’s conjecture to a cohomological condition on certain -equivariant vector bundles. In Section 5, we will check this condition in the setting of complex groups, thus verifying Vogan’s conjecture in a large family of cases.

## 2. Preliminaries

Fix as in Section 1. Write for the complexifications and for the (complex) Lie algebras. Let denote the Cartan involution corresponding to and let be the -eigenspace of . There is a Cartan decomposition .

A -module is a left module for the universal enveloping algebra of together with an algebraic -action such that

(1)

The action map is -equivariant,

(2)

The -action on , coming from the inclusion , coincides with the differentiated action of .

A morphism of -modules is a -module homomorphism which intertwines the -actions. Let denote the (abelian) category of -modules. A Harish-Chandra module is a -module which is finitely generated for . Note that an irreducible -module is automatically Harish-Chandra. Write for the full subcategory of Harish-Chandra modules.

### 2.1. Unipotent representations

Let denote the Langlands dual of , and write , for the nilpotent cones. The nilpotent orbits for and are related via Barbasch-Vogan duality (see Reference 4). This is a map

A nilpotent orbit is called special if it lies in the image of .

Every nilpotent -orbit gives rise to an infinitesimal character for as follows. If we fix a Cartan subalgebra , there is a Cartan subalgebra , which is canonically identified with . Using a -invariant identification , we can regard as a nilpotent -orbit in . Choose an element and an -triple . Conjugating by if necessary, we can arrange so that . Put

This element is well-defined modulo the (linear) action of the Weyl group and thus determines an infinitesimal character for by means of the Harish-Chandra isomorphism. Let be the unique maximal ideal with infinitesimal character . A special unipotent ideal is any ideal in which arises in this fashion.

Write for the associated variety of a two-sided ideal (i.e. the vanishing locus of ). By Reference 4, Prop A2, we have

In particular, the associated variety of a special unipotent ideal is (the closure of) a special nilpotent orbit (this explains the word ‘special’ in ‘special unipotent’).

Special unipotent ideals belong to a larger class of maximal ideals called simply unipotent ideals. This more general class of ideals is defined in Reference 14, Definition 6.0.1. If is a unipotent ideal, one can define the notion of a unipotent representation analogously to Definition 2.1. Our main results (Theorem 4.1 and Corollary 5.2) apply, without modification, to this more general class of representations (and the proofs are identical). Since we will not recall here the definition of this more general class of representations, the reader may choose to interpret ‘unipotent’ to mean ‘special unipotent’ wherever it is used below.

### 2.2. Associated varieties and associated $K$-cycles

Following Reference 18, we will associate to every Harish-Chandra module some geometric data in . We will need the concept of a good filtration of . A filtration of

by complex subspaces is compatible if

(1)

for every .

(2)

for every .

Under these conditions, has the structure of a graded, -equivariant -module. Our compatible filtration is good if

(3)

is finitely-generated over .

There is an equivalence of categories (obtained by taking global sections) between the category of -equivariant coherent sheaves on the affine space and the category of finitely-generated -equivariant -modules. Thus if is equipped with a good filtration, we can (and will) regard as an object in . The following is standard (see Reference 18, Sec 1 for a proof).

Proposition 2.2 provides a recipe for attaching geometric invariants to Harish-Chandra modules. A function with values in a semigroup is additive if whenever there is a short exact sequence . Under this condition, is well-defined on classes in and therefore (by Proposition 2.2) induces an (additive) function on Harish-Chandra modules.

The first example of this construction is the associated variety of a Harish-Chandra module . Let be the set of Zariski-closed subsets of with addition defined by . Let be the function

Since support is additive, induces an (additive) function on . If , we write for .

The next result relates the associated variety of to the associated variety of its annihilator.

In Section 3.2, we will give a simple proof of Theorem 2.3(iv) using the machinery of microlocalization.

In Reference 18, Vogan introduces a refinement of called the associated -cycle which carries additional information about the -action on . Consider the variety

The group acts on and, by (ii) of Theorem 2.3, with finitely many -orbits. Denote the -orbits on by

Note that if is a finite-length Harish-Chandra module, then by (iii) of Theorem 2.3

The set of associated -cycles forms a semigroup, with addition defined by

Now if , we can define an associated -cycle

where

The function can be extended to the category of -equivariant coherent sheaves set-theoretically supported in in the following manner: if choose a finite filtration by -equivariant subsheaves such that for every (for example, take for each ). Now define

In Reference 18, Thm 2.13, Vogan shows that is well-defined and additive. If is a finite-length Harish-Chandra module, then . So induces an (additive) function on finite-length Harish-Chandra modules , called the associated -cycle.

If is a special unipotent Harish-Chandra module, see Definition 2.1, then is of a very special form. Let be a -orbit. Let be the universal -equivariant cover and let be the canonical bundle on (i.e. the line bundle of top-degree differential forms).

This condition has a very simple Lie-theoretic interpretation. If we choose a point , there is an equivalence of categories

From left to right, the equivalence is given by restriction to the fiber over . A vector bundle is admissible if and only if the corresponding -representation satisfies

This is the ‘admissibility’ condition described in Reference 18, Sec 7. We note that if is a short exact sequence of -equivariant vector bundles on and , are admissible, then is admissible (this is evident from Equation 2.2). So admissibility is a property which can be ascribed to classes in .

Only certain nilpotent -orbits admit admissible vector bundles, so Theorem 2.6 imposes strong additional constraints on the associated varieties of special unipotent representations.

### 2.3. Vogan’s conjecture

In Reference 18, Sec 12, Vogan formulates a conjecture regarding the -structure of special unipotent Harish-Chandra modules, under some conditions.

In Section 4, we will show that Conjecture 2.7 is true if satisfies a certain cohomological condition, which we can verify in many cases.

If we choose a point and write for the centralizer of in , then corresponds to a finite-dimensional representation of and, as representations of , . Thus, the conclusion of Conjecture 1.1 follows from that of Conjecture 2.7.

### 2.4. Localization of categories

In this section, we will review some basic aspects of the theory of localization of abelian categories. The discussion here follows Reference 17, Chapter 4.

Let be an abelian category. A full subcategory is called Serre if for every short exact sequence in ,

In other words, is a full subcategory which is closed under the formation of subobjects, quotients, and extensions.

Given a Serre subcategory , one can define the quotient category . The next (very standard) result describes the universal property which it satisfies.

We are interested in Serre subcategories of a very special type. A Serre subcategory is localizing if the quotient functor admits a right-adjoint . We call the composition the localization of with respect to the localizing subcategory .

Proposition 2.12 catalogs the essential properties of .

We conclude this section with a useful criterion which we will use below.

## 3. Microlocalization of Harish-Chandra modules

### 3.1. The Rees construction

We want to perform operations on filtered Harish-Chandra modules analogous to the restriction and extension of coherent sheaves on . The first problem we encounter is that the category of Harish-Chandra modules with good filtrations (in short, ‘well-filtered Harish-Chandra modules’) is not abelian. Cokernels are not well-defined. The solution is to pass to a larger abelian category containing .

Let be an associative algebra equipped with an increasing filtration by subspaces

Form the algebra of Laurent series in the formal symbol . Define a -grading by declaring . The Rees algebra of is the graded subalgebra

In a precise sense, interpolates between