# Group Embeddings: Geometry and Representations (07w5034)

Arriving in Banff, Alberta Sunday, September 16 and departing Friday September 21, 2007

## Organizers

(Université de Grenoble)

(Loyola University Chicago)

(University of Western Ontario)

Ernest Vinberg (Moscow State University)

## Objectives

1. GROUP EMBEDDINGS AND SPHERICAL VARIETIES

The overarching goal of the proposed conference is to further the
discussion and understanding of group embeddings. Spherical varieties
[L-V] play a central role. These include torus embeddings [K-K-M-S],
reductive monoids [P-R], and symmetric varieties [D-P].

The "wonderful compactification" stands out among all others as the
one embedding of a semisimple group that serves as the role model for
further development. DeConcini and Procesi [D-P] originally
calculated the Betti numbers and cell decomposition using the method
of Bialinycki-Birula. Since then, important work has been done by
Brion [Br3] and Springer [Sp] on the geometry of orbit closures. Kato
[Ka1] and Tchoudjem [Tc] have found an analogue of the Borel-Weil-Bott
theorem, and Kato [Ka2] described all the equivariant vector bundles.
The wonderful compactification can also be described in terms
of certain reductive monoids [Re3]. Several authors have discovered direct
cell decompositions of the wonderful compactification [Br], [Ka1], [Re],
[Sp].

Another major objective is to assess and discuss more general classes
of embeddings that can be understood using what is known about the
wonderful compactification. In particular, to find cell
decompositions, Betti numbers, and decisive extensions of
Borel-Weil-Bott theory. And what can be said in positive
characteristics?

Regarding more general classes of embeddings, much has been done
recently, and needs to be discussed in the forum of a major
international conference. Here is a partial list of some of those key
topics:

a) Complexity of group actions [Ar], [Ti2].

b) Stable reductive varieties [A-B1], [A-B2].

c) Analogues of the Bruhat decomposition for spherical varieties [Br4],
[Re1], [Ti1].

d) Equivariant compactifications of spherical homogeneous spaces
[B-L-V], [Kn], [D-P].

e) Cartier and Weil divisors on spherical varieties [Br2], [Ti3].

f) Universal G_m torsors on certain moduli spaces [Co], [H-T.a].

g) Embeddings of $G_a^n$; an important beginning with many interesting
examples [H-T.b].

h) Compactification of Jacobian varieties [Al].

i) Among group embeddings and spherical varieties there should be many
opportunities to identify and study examples of "Cox rings" [Co].

j) Obtain a class of embeddings for nonreductive groups that is
well-behaved geometrically. Such embeddings will lead to
nonreductive groups.

k) Moduli spaces of group compactifications; generalizations of the
toric Hilbert scheme [Al], [A-B1], [A-B2], [A-B3], [H-S], [P-S].

Some problems of interest in this area include the following:

1) Describe the cohomology rings of smooth, complete group
embedddings, by generators and relations.

(For toric varieties, the answer is given by a result of Jurkiewicz
and Danilov. For "regular" group embeddings, the equivariant
cohomology ring has been described by Bifet, De Concini and Procesi
as a subring of a larger ring. But no generators of the cohomology
ring are known in general).

2) Construct B x B-equivariant desingularizations of the closures
of B x B-orbits in G-embeddings (G a connected reductive
group, B a Borel subgroup). Does there exist such a desingularization
with only finitely many fixed points of the maximal torus T x T ?

(This reduces easily to constructing a B x B-equivariant
desingularization of the closure of B; for regular embeddings,
this closure is almost always singular.

An affirmative answer to the second question implies that
the intersection cohomology of B x B-orbit closures vanishes
in all odd degrees. This was proved by Springer for the wonderful
compactification, by combinatorial arguments.)

3) Study the topology of hypersurfaces in smooth complete group
embeddings: determine their numerical invariants, by generalizing
the known results for toric varieties.

4) (Related to problem 1). In the case of a smooth, projective group
embedding X, find an explicit (canonical?) cell decomposition of
X, similar to what was done by Brion, Springer and Renner for the
wonderful compactification.

(How is each cell made up of B x B-orbits? Can one find the
appropriate Bialinycki-Birula 1-parameter subgroup? Is there a
monoid-theoretic way to do it in some cases?)

2. ALGEBRAIC MONOIDS

Another central goal of the conference is to discuss aspects of the
theory of algebraic monoids related to the geometry of embeddings, and
to further the understanding of representations of reductive monoids.
Reductive monoids are group embeddings of a particular kind. They can
be understood as reductive semigroups [P-R], and also as spherical
varieties [Rit]. Their structure is sufficiently rich so as to offer
hints into more general problems about spherical varieties [Rit].
Classification problems here will lead to birational geometry related
to the combinatorics of orbit closures, hyperplane arrangements, and
flat deformations of a semisimple group [Vi].

Monoids can also be used to produce elementary constructions that are
of geometric importance (eg. of cells). One can describe these cells
of the wonderful compactification explicitly in terms of B x B-orbits
[Re3] of a certain reductive monoid.

Another important topic is the topology of the adjoint quotient [Re2].
This interesting space has a canonical cell decomposition that does
not come about from a BB-decompostion. This space should be useful in
identifying new character formulas that arise from the Riemann-Roch
formalism.

Reductive monoids are plentiful. One can choose a finite dimensional
representation V of a reductive group G, then the Zariski closure in
End(V) of the group generated by the scalars and the image of G will
be an interesting reductive monoid [Sol]. A particularly interesting
class of examples is obtained by selecting the natural representation
of a classical group. These "classical" monoids were studied in
[Do1]. One would like to establish their (conjectured) normality.
There would be similar questions for exceptional types, and also for
other choices of representations. Are there general techniques from
spherical embeddings that can be applied to such examples?

Solomon has studied finite monoids (the Renner monoids), analogues of
the Weyl group of a reductive group. The simplest class of examples is
the so-called "rook monoid" which is a rather natural generalization
of the symmetric group. There is recent interesting work of Halverson
and Ram (see [Hal], {Hal-R]) on a q-analogue of the rook monoid. This
is just the tip of a large iceberg; analogous structures exist in
abundance, and little is known in general.

There are also some recent results on monoids related to Kac-Moody
groups [Mok].

3. REPRESENTATION THEORY

Another goal of the conference is to study the representation theory
of an arbitrary (reductive) monoid. This is a certain interesting
piece of the rational representation theory of its unit group, a
reductive algebraic group, and to distinguish that piece, one needs to
work out an appropriate combinatoric. There will be generalized Schur
algebras (in the sense of Donkin [Don]) implicit in the coordinate
bialgebra of the monoid, and the representation theory of the monoid
breaks up into a direct sum of the representation theories of the
various generalized Schur algebras, which are finite dimensional
quasihereditary algebras.

All of this extends the classic motivating example of polynomial
representation theory of general linear groups. Although this goes
back to Schur's dissertation, at the very inception of representation
theory as a mathematical discipline, it remains in positive
characteristic a vigorous and intensively studied aspect of modern
representation theory, with important work in the past twenty-five
years by Green, James, Donkin, Erdmann, Cline, Parshall, and Scott,
Dlab, Ringel, and others. There are connections with representation
theory of symmetric groups, and in the modular case some aspects of
these connections have been clarified. Still, some of the main
questions (e.g., characters of the modular representations, computing
cohomology and ext-groups) remain unanswered. How does the geometry of
group embeddings shed any light on these questions? Another
interesting open question is to describe the blocks of a given
reductive monoid. The correponding problem was settled by Donkin for
reductive groups, and there is a natural conjecture for the reductive
monoid case.

One may want to consider deformations of the coordinate bialgebra of
an algebraic monoid. These will lead naturally to generalized q-Schur
algebras, which have important applications to modular representations
of finite groups [D-J], [Ge]. One may also want to consider the dual
viewpoint, where the coordinate bialgebra is replaced by the
corresponding quantized enveloping algebra of the appropriate Lie
algebra. So there are important connections with the canonical bases of
Lusztig and Kashiwara, and all the associated combinatorial apparatus.
Generalizing Schur-Weyl duality in these contexts is an
interesting problem, related to the study of various Iwahori-Hecke
algebras, Birman-Wenzl algebras, and so forth. Work of Littelmann [Lit]
and Berenstein-Zelevinsky [B-Z] is related.

Another related topic is the structure and representation theory of a
finite monoid. Putcha [Pu] has recently shown that the complex monoid
algebra of a finite regular monoid is quasihereditary, thus providing
another important link to the theory of quasihereditary algebras.

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