 # On the Cohomology of Certain Non-Compact Shimura Varieties (AM-173)

Sophie Morel
with an appendix by Robert Kottwitz
Pages: 256
https://www.jstor.org/stable/j.ctt7tb2g

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-xii)
4. Chapter One The fixed point formula
(pp. 1-30)

The reference for this section is [P2] §3.

Let$\mathbb{S} = {R_{\mathbb{C}/\mathbb{R}}}{\mathbb{G}_{m,\,\mathbb{R}}}.$Identify$\mathbb{S}(\mathbb{C}) = {(\mathbb{C}\;{ \otimes _\mathbb{R}}\mathbb{C})^ \times }$and${\mathbb{C}^ \times }\; \times \;{\mathbb{C}^ \times }$using the morphism$a\; \otimes \;1\; + \;b\; \otimes \;i\; \mapsto (a\; + \;ib,\;a\; - \;ib),$and write${\mu _0}:{\mathbb{G}_{m,\,\mathbb{C}}} \to {\mathbb{S}_{\text{C}}}$for the morphism$z \mapsto (z,\;1)$.

The definition of (pure) Shimura data that will be used here is that of [P2] (3.1), up to condition (3.1.4). So a pure Shimura datum is a triple$({\mathbf{G}}{\text{,}}\;\mathcal{X},\;h)$(that will often be written simply$({\mathbf{G}}{\text{,}}\;\mathcal{X})$), where G is a connected reductive linear algebraic group over$\mathbb{Q}$,$\mathcal{X}$is a set with a transitive action of${{\mathbf{G}}_\mathbb{R}}$and$h:\;\mathcal{X}\;\xrightarrow{{}}\;{\text{Hom(}}\mathbb{S},\;{{\mathbf{G}}_\mathbb{R}})$is a${{\mathbf{G}}_\mathbb{R}}$-equivariant morphism, satisfying conditions (3.1.1), (3.1.2), (3.1.3), and (3.1.5) of [P2], but not necessarily...

5. Chapter Two The groups
(pp. 31-46)

In the next chapters, we will apply the fixed point formula to certain unitary groups over$\mathbb{Q}$. The goal of this chapter is to define these unitary groups and their Shimura data, and to recall the description of their parabolic subgroups and of their endoscopic groups.

For$n\; \in \;\mathbb{N}*$, write

$I\; = \;{I_n} = \left( {\begin{array}{*{20}{c}} 1 & {} & 0 \\ {} & \ddots & {} \\ 0 & {} & 1 \\ \end{array} } \right)\; \in \;{\mathbf{G}}{{\mathbf{L}}_n}(\mathbb{Z})$

and

${A_n} = \left( {\begin{array}{*{20}{c}} 0 & {} & 1 \\ {} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {} \\ 1 & {} & 0 \\ \end{array} } \right)\; \in \;{\mathbf{G}}{{\mathbf{L}}_n}(\mathbb{Z})$.

Let$E\; = \;\mathbb{Q}[\sqrt { - b} ]\;(b\; \in \;\mathbb{N}*\;{\text{square - free)}}$be an imaginary quadratic extension of$\mathbb{Q}$. The nontrivial automorphism ofEwill be denoted${\text{by}}\;\overline {\;\;}$. Fix once and for all an injection$E\; \subset \;\bar \mathbb{Q}\; \subset \;\mathbb{C},$and an injection$\bar \mathbb{Q}\; \subset \;{{\bar \mathbb{Q}}_p}$for every prime numberp.

Let$n\; \in \;\mathbb{N}*$and let$J\; \in \;{\mathbf{G}}{{\mathbf{L}}_n}(\mathbb{Q})$be a symmetric matrix. Define an algebraic group...

6. Chapter Three Discrete series
(pp. 47-62)

Let G be a connected reductive algebraic group over$\mathbb{R}$. In this chapter, we form theL-groups with the Weil group${W_\mathbb{R}}$. Remember that${W_\mathbb{R}} = {W_\mathbb{C}} \sqcup {W_\mathbb{C}}\tau$, with${W_\mathbb{C}} = {\mathbb{C}^ \times },$${\tau ^2} = - 1\; \in \;{\mathbb{C}^ \times }$and, for every$z\; \in \;{\mathbb{C}^ \times }$, and that$\tau z{\tau ^{ - 1}} = \bar z,$acts on$\widehat{\mathbf{G}}$via its quotient${\text{Gal(}}\mathbb{C}{\text{/}}\mathbb{R}{\text{)}}\; \simeq \;{W_\mathbb{R}}/{W_\mathbb{C}}.$. Let$\Pi ({\mathbf{G}}{\text{(}}\mathbb{R}{\text{))}}$(resp.,${\Pi _{{\text{temp}}}}({\mathbf{G}}{\text{(}}\mathbb{R}{\text{))}}$) be the set of equivalence classes of irreductible (resp., irreducible and tempered) admissible representations of${\mathbf{G}}{\text{(}}\mathbb{R}{\text{)}}$. For every$\pi \; \in \;\Pi ({\mathbf{G}}{\text{(}}\mathbb{R}{\text{))}}$, let${\Theta _\pi }$be the Harish-Chandra character of$\pi$(seen as a real analytic function on the set${{\mathbf{G}}_{{\text{reg}}}}(\mathbb{R})$of regular elements of${\mathbf{G}}{\text{(}}\mathbb{R}{\text{))}}$.

Assume that${\mathbf{G}}{\text{(}}\mathbb{R}{\text{)}}$has a discrete series. Let${{\mathbf{A}}_G}$be the maximal...

7. Chapter Four Orbital integrals at p
(pp. 63-78)

Lemma 4.1.1(cf. [K3] 2.1.2, [K9] p. 193) Let F be a local or global field and${\text{G}}$be a connected reductive algebraic group over F. For every cocharacter$\mu :\;{\mathbb{G}_{m,\,F}}\xrightarrow{{}}\;{\mathbf{G}},$there exists a representation${r_\mu }$of${}^L{\mathbf{G}}{\text{(}} = \widehat{\mathbf{G}}\; \rtimes \;{W_F}),$unique up to isomorphism, satisfying the following conditions:

(a) The restriction of${r_\mu }$to$\widehat{\mathbf{G}}$is irreducible algebraic of highest weight$\mu$.

(b) For every${\text{Gal(}}\bar F\,/\,F)$-fixed splitting of$\widehat{\mathbf{G}}$, the group${W_F}$, embedded in${}^L{\mathbf{G}}$by the section associated to the splitting, acts trivially on the highest weight subspace of${r_\mu }$(determined by the same splitting).

Letpbe a prime number,${{\bar \mathbb{Q}}_p}$...

8. Chapter Five The geometric side of the stable trace formula
(pp. 79-84)

We use the following rules to normalize the Haar measures.

(1) In the situation of theorem 1.6.1, use the normalizations of this theorem.

(2) Let G be a connected reductive group over$\mathbb{Q}$. We always take Haar measures on${\mathbf{G}}({\mathbb{A}_f})$such that the volumes of open compact subgroups are rational numbers. Letpbe a prime number such that G is unramified over${\mathbb{Q}_p},$and letLbe a finite unramified extension of${\mathbb{Q}_p}$; then we use the Haar measure on${\mathbf{G}}(L)$such that the volume of hyperspecial maximal compact subgroups is 1. If a Haar measure$d{g_f}$on${\mathbf{G}}({\mathbb{A}_f})$...

9. Chapter Six Stabilization of the fixed point formula
(pp. 85-98)

To simplify the notation, we suppose in this chapter that the group G is${\mathbf{GU}}{\text{(}}p,\;q)$, but all the results generalize in an obvious way to the groups${\mathbf{G}}{\text{(}}{\mathbf{U}}{\text{(}}{p_1},\;{q_1})\; \times \; \cdots \; \times \;{\mathbf{U}}{\text{(}}{p_r},\;{q_r}))$.

We first rewrite the fixed point formula using proposition 3.4.1.

The notation is as in chapters 1 (especially section 1.7), 2 and 3. Fix$p,\;q\; \in \;\mathbb{N}$such that$n: = p\; + \;q\; \geqslant \;1,$and let${\mathbf{G}} = {\mathbf{GU}}{\text{(}}p,\;q)$. We may and will assume that$p\; \geqslant \;q$. LetVbe an irreducible algebraic representation of${{\mathbf{G}}_\mathbb{C}}$and$\varphi :\;{W_\mathbb{R}}\xrightarrow{{}}\;{}^L{\mathbf{G}}$be an elliptic Langlands parameter corresponding to$V*$as in proposition 3.4.1. Let$K\; \subset \;\mathbb{C}$be a number field such thatVis defined...

10. Chapter Seven Applications
(pp. 99-118)

This chapter contains a few applications of corollary 6.3.2. First we show how corollary 6.3.2 implies a variant of theorem 5.4.1 for the unitary groups of section 2.1. The only reason we do this is to make the other applications in this chapter logically independent of the unpublished [K13] (this independence is of course only formal, as the whole stabilization of the fixed point formula in this book was inspired by [K13]). Then we give applications to the calculation of the (Hecke) isotypical components of the intersection cohomology and to the Ramanujan-Petersson conjecture.

The simplest way to apply corollary 6.3.2...

11. Chapter Eight The twisted trace formula
(pp. 119-156)

We first recall some definitions from section 1 of [A4].

Let${{\mathbf{\tilde G}}}$be a reductive group (not necessarily connnected) over a fieldK. Fix a connected component G of${{\mathbf{\tilde G}}}$, and assume that G generates${{\mathbf{\tilde G}}}$and that${\mathbf{G}}{\text{(}}K)\; \ne \;\emptyset$. Let${{\mathbf{G}}^0}$be the connected component of${{\mathbf{\tilde G}}}$that contains 1.

Consider the polynomial

${\text{det((}}t\; + \;1)\; - \;{\text{Ad(}}g),\;{\text{Lie(}}{{\mathbf{G}}^0}))\; = \;\sum\limits_{k \geqslant 0} {{D_k}(g){t^k}}$

on${\mathbf{G}}{\text{(}}K)$. The smallest integerkfor which${D_k}$does not vanish identically is called therankof G; we will denote byr. An elementgof${\mathbf{G}}{\text{(}}K)$is calledregularif${D_r}(g) \ne 0$.

Aparabolic subgroupof${{\mathbf{\tilde G}}}$is the normalizer in${{\mathbf{\tilde G}}}$of a...

12. Chapter Nine The twisted fundamental lemma
(pp. 157-188)

The goal of this chapter is to show that, for the special kind of twisted endoscopic transfer that appears in the stabilization of the fixed point formula and for the groups considered in this book (and some others), the twisted fundamental lemma for the whole Hecke algebra follows from the twisted fundamental lemma for the unit element. No attempt has been made to prove a general result, and the method is absolutely not original: it is simply an adaptation of the method used by Hales in the untwisted case ([H2]), and this was inspired by the method used by Clozel...

13. Appendix Comparison of two versions of twisted transfer factors
(pp. 189-206)
Robert Kottwitz
14. Bibliography
(pp. 207-214)
15. Index
(pp. 215-218)