Characters of unipotent groups over finite fields
Topic
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)
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)
Speakers
Additional Information
Algebraic Geometry Seminar 2007
Dmitriy Boyarchenko (University of Chicago)
Dmitriy Boyarchenko (University of Chicago)