Representation theory of the finite unipotent linear groups

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REPRESENTATION THEORY OF THE FINITE UNIPOTENT LINEAR GROUPS

Ning Yan

A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy 2001

Supervisor of Dissertation

Graduate Group Chairperson

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UMI Number: 3015396

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ACKNOWLEDEMENTS

I would like to express deep gratitude to my advisor, Alexandre Kirillov, for his guidance during this work. I am also deeply grateful to Michael Larsen for his earlier help and support.

I thank the Department of Mathematics at the University of Pennsylvania for its friendliness and patience. In particular, I would like to thank the following faculty members: Jonathan Block, Ching-Li Chai, Ted Chinburg, Chris Croke, Dennis DeTurck, Ron Donagi, Charles Epstein, Murray Gerstenhaber, Herman Gluck, David Harbater, Richard Kadison, Tony Pantev, Mary Pugh, Steve Shatz, Harry Tamvakis, Herbert Wilf, Jianqiang Zhao and Wolfgang Ziller. Each of them has been, in one way or another, a source of inspiration to me. Finally, I cannot thank Janet Burns enough for taking care of me throughout the years.

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ABSTRACT

REPRESENTATION THEORY OF THE FINITE UNIPOTENT LINEAR GROUPS Ning Yan Alexandre Kirillov

The general linear group G L(n, K) over a field K contains a particularly prom inent-, subgroup, the uni-potent linear group £/X(n,K), consisting of all the unipotent upper triangular elements. In this paper we are interested in the case when K is the finite field F,, and our goal is to better understand the representation theory of UL(n, F9). The complete classification of the complex irreducible representations of this group has long been known to be a difficult task. The traditional orbit method, famous for its success when K has characteristic 0 , is a natural source of intuition and conjectures, but in our case the relation between coadjoint orbits and complex representations is still a mystery. Here we construct and study a subring in the representation ring of UL(n,Fq), and build a theory with many of the major features one would expect from the philosophy of orbit method.

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CONTENTS

Introduction

Si.

Basic notations

§2 .

Transition characters

§3.

Transition orbits and cotransition orbits

§4.

The number of transition characters

§5.

The degree and the self-interwining number

§6 .

The primary decomposition

§7.

Decomposition of tensor products

C00 00

§0 .

The discrete series character

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Introduction In a recent paper A.A. Kirillov wrote [17]: “The set of triangular matrices is one of the most fundamental objects in mathematics, as well as the set of natural numbers or the group of permutations”. A corollary of this metamathematical statement is that the group we are about to discuss in the present work—the group of unipotent upper triangular matrices —deserves an appropriate name, which it does not seem to have acquired so far. Since this group is truely one of the most important subgroups of the general linear group (especially when it comes to representation theory), we propose to bestow on it the title unipotent linear group, even at the risk of introducing some level of ambiguity, as the title might also be understood as referring to one of its algebraic subgroups (or in other words, one of the unipotent algebraic linear groups). We further propose the notation U L(n,K ) for the unipotent linear group over a field K. The main purpose of this work is to improve our understanding of UL(n) over a finite field. Here we are dealing with a finite group, our ultimate goal is to build its representation theory with explicit constructions and a complete classification of the complex irreducible representations, similar in spirit to the representation theory of the symmetric group. The philosophy of orbit method plays a significant role in this ongoing project. The history goes back to the early 1960’s, when Kirillov [14] introduced his orbit method into the representation theory of nilpotent Lie groups. An ideal example

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where this method is most successful is the group UL(n, R). In the 1970’s the orbit method was adapted, with some restrictive assumptions, to the case over a finite field by D. Kazhdan [13]. This adaption applies in particular to UL(n,Fq) when F, has large enough characteristic. The group UL(n,Fq) was recently taken up anew by Kirillov, who made a conjecture in the form of a character formula [17]. It turned out that the conjecture was overly optimistic, as computer calculations [12] soon provided (indirect) evidence that in fact the character values should not always lie in the cyclotomic field suggested by the conjectured formula. The situation became all the more tantalizing when all attempts at producing a character which directly contradicts the conjecture had failed. In any case, the “pseudo-characters” (which are in one-to-one correspondence with the coadjoint orbits) offered by Kirillov’s conjecture form a most natural orthonormal basis for the space of complex-valued class functions on the group, and it is an extremely intriguing problem to find the transition matrix which connects this basis to the actual irreducible characters. We like to mention here that the classification of the coadjoint orbits offers another nice challenge. The present work starts with two simple observations. First, the coadjoint action of UL(n, F?) on the dual space of its Lie algebra can be extended to a double action, the cotransition action. Second, the group algebra of UL(n,Wq) has a natural basis, the Fourier basis. The cotransition action and the Fourier basis are connected through the regular representation of UL(n,Fq) on its group algebra, and this connection leads to a rich theory of transition characters with many of the major features one

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would expect from the philosophy of orbit method. These features include an oneto-one correspondence between the transition characters and the cotransition orbits, a character formula expressing the character in terms of the orbit, and equivalence between character operations (tensor product, induction and restriction) and orbit operations. A remarkable new feature is the canonical decomposition of the regular representation into a direct sum of transition modules. The collection of all transition characters form an orthogonal basis for the C-linear space of transition-invariant functions on UL(n,Fq). The collection of all Z-linear combinations of transition characters form a subring in the representation ring of UL(n,Fq). This subring is generated by the primary transition characters, and each transition character has a unique primary decomposition into a tensor product of primary transition characters. The last part of this work was inspired by a paper of G.I. Lehrer [19]. Lehrer was interested in the discrete series character of t/X(n, F,), which is by definition the common restriction of the discrete series characters of GL(n, F9). He derived a tensor product decomposition of this character, and then further decomposed each factor into a multiplicity-free sum of irreducible components. We noticed that these irreducible components are precisely the primary transition characters of our theory. This led us to the discovery of an explicit decomposition of the discrete series character into a sum of transition characters. Moreover, we were able to construct a model of the

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discrete series representation as a canonical submodule in the regular representation of U L(n,¥q). To close this introduction we draw the reader’s attention to another previous work on the representation theory of UL(n, F,), by C.A.M. Andre [1,2,3,4]- Throughout his papers Andre makes the assumption p > n in order to have the benefit of a wellbehaved exponential map. In this case an one-to-one correspondence between the irreducible characters and the coadjoint orbits was established by Kazhdan. Andre defines certain “basic sum s” of coadjoint orbits, and studies the characters associated to the orbit sums via the above correspondence. Note that here the popular mode of thinking is to go from F, to its algebraic closure, so that the coadjoint orbits can be treated as geometric objects. It transpired that the orbit sums of Andre are precisely the cotransition orbits, and the characters in his work are the transition characters. The main results of Andre agree well with our own theory, while his approach is an impressive tour de force in algebraic geometry.

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1

Basic notations

Let uI(n,K) be the K-algebra of nilpotent upper triangular matrices over K. Let ut*(n, K) be the dual K-linear space of u((n, K). An element in u f(n , K) is a K-linear map from ul(n, K) to K. There are three natural ways for UL(n, K) to act on u[(n,K). These are defined by: left transition

X

>—► g • X

right transition

X

i—*■ X ■g

adjoint action

X

g • X • g~l

where g € UL(n, K) and X 6 ul(n, K). The dot (•) means multiplication of matrices. The left and right transitions commute, and together they give a double action of UL(n, K) on ut(n,K). We will refer to the orbits under this double action as the transition orbits. The adjoint orbits are the orbits under the adjoint action. Each transition orbit is a union of adjoint orbits. Correspondingly we have the following actions of UL(n, K) on u[*(n, K):

left

cotransition

(g*X)(X) = X(X • g)

right

cotransition

(X*g)(X) = X(g ■X )

coadjoint action

X9(X)

=

A(