## Characters of unipotent groups over finite fields

- Date: 03/12/2007

Dmitriy Boyarchenko (University of Chicago)

University of British Columbia

Let G be a connected unipotent group over a finite field

F_q. For each natural number n, we have the unique extension F_{q^n} of

F_q of degree n, and we can form the finite group G(F_{q^n}) of points

of G defined over F_{q^n}. An interesting problem, motivated by

Lusztig's theory of character sheaves, is to study irreducible

characters of these finite groups (over an algebraically closed field

of characteristic 0) and relate them to the geometry of G. If the

nilpotence class of G is less than p (the characteristic of the field

F_q), there exists an explicit description of irreducible characters of

G(F_{q^n}), provided by Kirillov's orbit method. It allows one to

introduce the notion of an L-packet of irreducible representations of

G(F_{q^n}). This notion is morally analogous to the notion of an

L-packet in Lusztig's theory, even though Lusztig's definition cannot

be applied to unipotent groups. If the nilpotence class of G is at

least p, no analogue of the orbit method is known to us. Nevertheless,

we have succeeded in giving a geometric definition of L-packets of

irreducible characters of G(F_{q^n}) for every connected unipotent

group G over F_q. My talk will be devoted to giving a precise statement

of our result, explaining some motivation behind it, and sketching a

few of the essential ideas used in its proof. (Morris): It is known

that finite-index subgroups of the arithmetic group SL(3,Z) are not

left orderable. (In other words, they have no interesting actions on

the real line.) This naturally led to the conjecture that most other

arithmetic groups (of higher real rank) also are not left orderable.

The problem remains open, but joint work with Lucy Lifschitz verifies

the conjecture for many examples, including every finite-index subgroup

of SL(2,Z[sqrt(3)]) or SL(2,Z[1/3]). The proofs are based on the fact,

proved by D.Carter, G.Keller, and E.Paige, that every element of these

groups is a product of a bounded number of elementary matrices.

(Joint work with Vladimir Drinfeld)

Algebraic Geometry Seminar 2007