**Title:** Characters of unipotent groups over finite fields (Joint work with Vladimir Drinfeld)

**Speaker:** Dmitriy Boyarchenko

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

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.

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