This is joint work with Dave Benson. Let $\mathfrak g$ be a finite dimensional nilpotent p-restricted Lie algebra over a field of characteristic p. For $p \geq 5$, we show that every endotrivial $\mathfrak g$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $p \geq 5$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.

We will discuss new bounds on character values for finite groups of Lie type, obtained in recent work of the speaker and collaborators. Some applications of these character bounds will be also described.

When the base ring is not a field, power reductivity of a group scheme is a basic notion, intimately tied with finite generation of subrings of invariants. Geometric reductivity is weaker and less pertinent in this context. We give a survey of these properties and their connections.

Much of this lecture will focus on the special case of elementary abelian $p$-groups, the ``source" of most work on support varieties. This connects the work of Brian Parshall, as well as Jon Carlson, Leonard Scott, and others to our recent work which is described below. For a finite group $\tau$ and a field $k$ of characteristic $p$ dividing the order of $\tau$, we construct a map $p_\tau: \bU_\tau \ \to \ \bP Y_\tau$ of varieties over $k$ with unions of affine spaces as fibers and with $(\bP Y_\tau)/\tau$ $p$-isogenous to $\Proj H^\bu(\tau,k)$. On $\bU_\tau$, we construct a ``universal $p$-nilpotent operator" $\bP \Theta_\tau$ which leads to the construction of $\tau$-equivariant coherent sheaves associated to finite dimensional $k\tau$-modules $M$. These coherent sheaves are algebraic vector bundles on $\bU_\tau$ if $M$ has constant Jordan type. For any finite dimensional $k\tau$-module $M$, these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of $M$. For a linear algebraic group $\bG$, we compare these constructions for the finite group $\bG(\bF_p)$ to previous constructions for the infinitesimal group scheme $\bG_{(r)}$. For a rational $\bG$-module $M$ of exponential type $< p^r$, we obtain a relationship between classes in algebraic $K$-groups for $\bG(\bF_p)$ and for $\bG_{(r)}$. The special case in which $\tau$ is an elementary abelian $p$-group serves as our guide for more general groups as well as extends the vector bundle invariants earlier obtained for such groups.

I will discuss motivic representations of finite groups of Lie type arising from Deligne-Lusztig varieties and the relation with rationality properties of representations.

Hecke endomorphism algebras are a natural generalization of q-Schur algebras from symmetric groups to arbitrary Coxeter groups. They appear naturally in the study of representations of finite groups of Lie type, especially finite general linear groups whose representations are connected with those of quantum general linear groups. Generally speaking, the structure and representations of Hecke endomorphism algebras are difficult to understand if the associated Coxeter group is not the symmetric group. Over twenty years ago, B. Parshall, L. Scott and the speaker investigated some rough stratification structure for those associated with Weyl groups. We conjectured the existence of a finer stratification by Kazhdan--Lusztig two-sided cells for an enlarged endomorphism algebra. I will report on the progression of ideas in our successful efforts to prove a (very slightly modified) version of the conjecture. Inspired by an Ext${}^1$ vanishing condition uncovered in a local case, we use exact categories to formulate by analogy a tractable global version, not mentioning localization and often requiring less vanishing. After constructing many relevant exact category settings, we are eventually able to prove this exact category Ext${}^1$ vanishing in one of them that contains all the filtered objects we need. If time permits, I will mention possible applications to representations of finite groups of Lie type and i-quantum algebras.

A Frobenius algebra is a (not necessarily associative) algebra that admits a nondegenerate invariant (associative) symmetric bilinear form. A fused algebra is a generalization of a graded algebra, a canonical example being a Jordan algebra equipped with its Peirce decompositions. But the motivating example of a fused Frobenius algebra is the real Griess algebra of dimension 196884 that is embedded in the Monster vertex operator algebra. Since Bob Griess' construction of the Monster in 1982, his algebra has been modified and generalized in many ways. We discuss a large class of generalizations of recent interest. In addition to Jordan algebras, this includes Majorana, axial, and decomposition algebras.

Let $X$ be the flag manifold of a simply connected semisimple group $G$ over $\mathbb C$. Let $K$ be a semifield (a structure in which addition, multiplication, division, but not substraction is defined). For example, K could be $\mathbb R_{>0}$, the strictly positive real numbers. It turns out that there is a form $G(K),X(K)$ of $G,X$ with $G(K)$ being a semigroup acting on $X(K)$. When $K=\mathbb R_{>0}$ this has been known earlier but for general $K$ this is new.