$p\text{-adic}$ analog of the Kazhdan-Lusztig Hypothesis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 15 April 2014

Notes and References

This is an excerpt of the paper $p\text{-adic}$ analog of the Kazhdan-Lusztig Hypothesis by A. V. Zelevinskii. Terrestrial Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 15, No. 2, pp. 9-21, April-June, 1981. Original article submitted November 27, 1980.

Introduction

In [Zel1980] the author has obtained a classification of irreducible complex representations of the groups ${GL}_n$ over a $𝔭\text{-adic}$ field $F$ (the results of [Zel1980] were announced in [Zel1977, Zel1977-2]). Let us briefly recall this classification. Irreducible representations of the groups $G_n=GL(n,F)$ are parametrized by collections of segments in the set of the cuspidal representations (the exact definitions are given below in Sec. 1). With each collection $a$ of segments we associate the induced module ${\pi }_a$ and the irreducible module $⟨a⟩$ that is the only irreducible submodule of ${\pi }_a\text{.}$ Particular cases of the representations ${\pi }_a$ are the representations that are induced by the one-dimensional representations of parabolic subgroups (in particular, representations of the principal series).

At present the computation of the multiplicity $m_{b,a}$ with which the irreducible representation $⟨b⟩$ occurs in the Jordan-Hölder series of the module ${\pi }_a$ is an open problem. Analogous problems arise in the representation theory in many other situations. In particular, there exist deep analogies between our problems and the problems on the computation of multiplicities of irreducible $𝔤\text{-modules}$ in the Verma module, where $𝔤$ is a complex semisimple Lie algebra.

It was not clear until recently as to in what terms the answer for multiplicities in the Verma modules must be expressed. The right language was found in [KLu1979, KLu1980] by Kazhdan and Lusztig. We describe the content of these papers that is interesting from our point of view.

Let $𝔥$ be a Cartan subalgebra of $𝔤,$ and $W$ be the Weyl group of the algebra $𝔤$ with respect to the sub-algebra $𝔥\text{.}$ To each character $\chi \in {𝔥}^{*}$ there correspond the Verma module $M_{\chi }$ and the irreducible module $L_{\chi }$ (see [Dix1977]). The modules $M_{\chi }$ have finite length and all of their composition factors have the form $L_{\psi }$ $\text{(}\psi \in {𝔥}^{*}\text{).}$ The problem is to compute the multiplicity $m_{\psi ,\chi }$ occurrence of $L_{\psi }$ in the Jordan-Hölder series of $M_{\chi }\text{.}$ The most interesting case is $\psi =w{\chi }_0,$ $\chi =v{\chi }_0,$ where $v,w\in W,$ and ${\chi }_0$ is an integral regular dominant weight. We know that the multiplicity $m_{\psi ,\chi }$ does not depend on ${\chi }_0$ (see [BGG1976]); we denote it by $m_{w,v}\text{.}$ Thus it is required to compute the function $m_{w,v}$ from a pair of elements of the Weyl group.

Let $\le$ denote the usual partial order relation on $W$ (see [Dix1977], 7.7).$†$ We know that $m_{w,n}>0$ if and only if $w\le v\text{;}$ moreover, $m_{w,w}=1$ for all $w\in W\text{.}$ In [KLu1979] the polynomial $P_{v,w}\left(q\right)=\mathbb{Z}\left[q\right]$ has been defined for each pair $w\le v$ and the hypothesis that its value for $q=1$ is equal to $m_{w,v}$ has been advanced. The polynomials $P_{v,w}$ satisfy recurrence relations so that they can be explicitly computed (see [KLu1979, Eq. (2.2.c]). Thus, we can assume the problem of computation of the coefficients $m_{w,v}$ as solved if the equality $m_{w,v}=P_{v,w}\left(1\right)$ is proved.

In [KLu1980] a cohomological interpretation of the polynomials $P_{v,w}$ is given in terms of the new cohomology theory of Goresky and MacPherson [GMa1968] (more precisely, in terms of its algebraic version, proposed by Deligne). This theory associates with each algebraic variety $X$ over an algebraically closed field $K$ of arbitrary characteristic the groups $H^i\left(X\right),$ which coincide with the usual $\text{(}l\text{-adic)}$ cohomologies for nonsingular $X$ and satisfy the Poincare duality for all $X\text{.}$ In Deligne's construction, a complex of sheaves $\pi$ is constructed on $X,$ and $H^i\left(X\right)$ are defined as the cohomologies of $X$ with coefficients in $\pi$ (if char $K=p,$ then $\pi$ consists of $l\text{-adic}$ sheaves, where $l\ne p\text{;}$ but if $K=ℂ,$ then $\pi$ can also be assumed to be complex). Let ${\mathscr{H}}^i\left(X\right)$ be the cohomological sheaves of the complex $\pi$ and for $x\in X$ let ${\mathscr{H}}^i{\left(X\right)}_x$ denote the stalk of the sheaf ${\mathscr{H}}^i\left(X\right)$ at the point $x\text{.}$ See [KLu1980] (see also Sec. 3 below) for the definition and the main properties of the sheaves $\pi$ and ${\mathscr{H}}^i\left(X\right)\text{.}$

Let $G$ be the complex semisimple algebraic group corresponding to $𝔤,$ and let $B$ be its Borel subgroup. By virtue of the Bruhat decomposition, the variety $G/B$ decomposes into the union of the Schubert cells $\underset{w\in W}{⨆}{X}_w,$ where $X_W$ is the $B\text{-orbit}$ of the point $x_w=wB\in G/B\text{.}$ We know that the closure ${\bar{X}}_W$ of the cell $X_W$ (it makes no difference whether the closure is taken in the complex topology or in the Zariski topology) is equal to $\underset{v\ge w}{⨆}{X}_v$
(see, e.g., [Ste1968, Sec. 8]).

Now, we are in a position to formulate the main result [KLu1980]: For $v\ge w$ the sheaf ${\mathscr{H}}^i\left({\bar{X}}_w\right)$ is equal to $0$ for odd $i,$ and $$P_{v,w}\left(q\right)=\sum _i\text{dim}\hspace{0.17em}{\mathscr{H}}^{2i}{\left({\bar{X}}_w\right)}_{x_v}·q^i\text{.}$$ In particular, the Kazhdan-Lusztig hypothesis takes the following form: $$\begin{array}{cc}{m}_{w,v}=\sum _i\text{dim}{\mathscr{H}}^{2i}{\left({\bar{X}}_w\right)}_{x_v}\text{.}& \text{(}*\text{)}\end{array}$$

The main result of the present article is a (hypothetical) expression for the multiplicities $m_{b,a},$ analogous to $\text{(}*\text{).}$ In this connection, the role of the variety $G/B$ is played by the variety $E=E\left(V\right)$ that consists of all linear operators of degree $+1$ acting in a fixed graded finite-dimensional vector space $V$ over the field $K\text{.}$ The automorphism group $\text{Aut}\hspace{0.17em}V$ of $V,$ preserving the graduation, acts naturally on $E,$ and its orbits on $E$ are the analogs of the Schubert cells. It is easily seen that they are enumerated by the collection of segments in $\mathbb{Z}\text{;}$ let us denote the orbit corresponding to a collection $a$ by $X_a\text{.}$ The connection of the varieties $X_a$ with the irreducible representations of the groups $G_n$ is very natural from the point of view of the Langlands reciprocity law (see [Zel1980, Sec. 10]).

Hypothesis. All sheaves ${\mathscr{H}}^i\left({\bar{X}}_b\right)$ are equal to $0$ for odd $i,$ and $$m_{b,a}=\sum _i\text{dim}{\mathscr{H}}^{2i}{\left({\bar{X}}_b\right)}_{x_a},$$ where $x_a$ is an arbitrary point of $X_a\text{.}$

In particular, this hypothesis implies that the multiplicity $m_{b,a}$ is nonzero if and only if $X_a\subset {\bar{X}}_b\text{.}$ This statement is proved in Sec. 2 of this article. To prove it we develop a combinatorial technique connected with collections of segments in $\mathbb{Z}\text{.}$ Their properties are analogous to the properties of the ordinary partitions; emphasizing this analogy, we call collections of segments in $\mathbb{Z}$ graded partitions. We study a partial order relation on graded partitions $\text{(}b\le a$ if $X_a\subset {\bar{X}}_b\text{);}$ it is the analog of the well-known relation of domination on partitions (see [LVi1973]).

In Sec. 3 our hypothesis is verified in all the cases for which the multiplicities $m_{b,a}$ are known (see [Zel1980, Sec. 11]). In all these cases we compute in a very simple manner the sheaves ${\mathscr{H}}^i\left({\bar{X}}_b\right),$ starting from the explicit construction of the resolution of singularities of the variety ${\bar{X}}_b\text{.}$ In particular, we compute the sheaves ${\mathscr{H}}^i$ on determinant varieties (i.e., on varieties of rectangular matrices whose ranks do not exceed a preassigned value). At present we have not been able to compute these sheaves for all ${\bar{X}}_b\text{;}$ in Sec. 3 we overcome the encountered difficulties.

Quite recently the Kazhdan—Lusztig hypothesis has been proved by J.-L. Brylinski and M. Kashiwara and, independently, by A. A. Beilinson and I. N. Bernshtein. In their works they have done a lot more; viz., they have constructed functors that give equivalence of a certain category of $𝔤\text{-modules}$ with a category of certain sheaves on $G/B\text{.}$ With the help of these functors, many notions of the representation theory are reformulated in terms of the variety $G/B\text{.}$ These functors have not yet been constructed in the $p\text{-adic}$ case (although there is no doubt that they do exist!).

In correspondence with this ideology, notions of the representation theory of the groups $G_n$ must be translated into the language of the varieties $X_b\text{.}$ In Sec. 4 we consider an interesting example of this type. In Sec. 9 of [Zel1980] an involution $\omega \to {\omega }^t$ has been constructed on the set of irreducible representations of the groups $G_n$ (it is interesting in that it interchanges the positions of the identity representation and the Steinberg representation). Obviously, the existence of such an involution has a general character; recently, it has been constructed for all finite Chevalley groups (see [Cur1980, Alv1979]). Our involution gives an involution on the set of graded partitions. Its description in the combinatorial terminology is not known (it must be analogous to the operation of transposition of the Young diagrams). In Sec. 4 we give a (hypothetical) description in the terminology of the linear algebra, i.e., we construct an involution on the cells of $X_b\text{;}$ this construction is a particular case of a general result of Pyasetskii [Pya1975].

The author is deeply thankful to A. A. Beilinson, I. N. Bernshtein, and P. Deligne for useful and fascinating discussions that significantly advanced this work. Thus, as a result of these discussions, the author understood as to how to compute the sheaves ${\mathscr{H}}^i$ on the determinant varieties; in addition, I. N. Bernshtein indicated [13] to the author. The author is also grateful to A. A. Beilinson and I. N. Bernshtein for acquainting him with the results of their article before its publication.

$†$ Often (see, e.g., [KLu1979, KLu1980]) $\le$ denotes the opposite relation. Under our choice, the identity element $e\in W$ is a maximal element.

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