887 research outputs found
On characters of Chevalley groups vanishing at the non-semisimple elements
Let G be a finite simple group of Lie type. In this paper we study characters
of G that vanish at the non-semisimple elements and whose degree is equal to
the order of a maximal unipotent subgroup of G. Such characters can be viewed
as a natural generalization of the Steinberg character. For groups G of small
rank we also determine the characters of this degree vanishing only at the
non-identity unipotent elements.Comment: Dedicated to Lino Di Martino on the occasion of his 65th birthda
Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
Motivated by string theory scattering amplitudes that are invariant under a
discrete U-duality, we study Fourier coefficients of Eisenstein series on
Kac-Moody groups. In particular, we analyse the Eisenstein series on ,
and corresponding to certain degenerate principal
series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show
that these Eisenstein series have very simple Fourier coefficients as expected
for their role as supersymmetric contributions to the higher derivative
couplings and coming from 1/2-BPS and 1/4-BPS
instantons, respectively. This suggests that there exist minimal and
next-to-minimal unipotent automorphic representations of the associated
Kac-Moody groups to which these special Eisenstein series are attached. We also
provide complete explicit expressions for degenerate Whittaker vectors of
minimal Eisenstein series on , and that have not
appeared in the literature before.Comment: 62 pages. Journal versio
Products of conjugacy classes and fixed point spaces
We prove several results on products of conjugacy classes in finite simple
groups. The first result is that there always exists a uniform generating
triple. This result and other ideas are used to solve a 1966 conjecture of
Peter Neumann about the existence of elements in an irreducible linear group
with small fixed space. We also show that there always exist two conjugacy
classes in a finite non-abelian simple group whose product contains every
nontrivial element of the group. We use this to show that every element in a
non-abelian finite simple group can be written as a product of two rth powers
for any prime power r (in particular, a product of two squares).Comment: 44 page
Some Problems in the Representation Theory of Simple Modular Lie Algebras
The finite-dimensional restricted simple Lie algebras of characteristic p > 5
are classical or of Cartan type. The classical algebras are analogues of the
simple complex Lie algebras and have a well-advanced representation theory with
important connections to Kazhdan-Lusztig theory, quantum groups at roots of
unity, and the representation theory of algebraic groups. We survey progress
that has been made towards developing a representation theory for the
restricted simple Cartan-type Lie algebras, discuss comparable results in the
classical case, formulate a couple of conjectures, and pose a dozen open
problems for further study.Comment: References updated; a few minor changes made in this versio
- …