# Group Embeddings: Geometry and Representations (07w5034)

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

## Organizers

Michel Brion (Institut Fourier)

Stephen Doty (Loyola University Chicago)

Lex Renner (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

numerical/cohomological information about representations of

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.

BIBLIOGRAPHY

[Al] Alexeev, V., Compactified Jacobians, ArXiv, 1996.

[A-B1] Alexeev, V., Brion, M., Stable reductive varieties I, Affine

varieties, Inventiones Math. 157 (2004), 227--274.

[A-B2] Alexeev, V., Brion, M., Stable reductive varieties II, Projective

varieties, Adv. Math. 184 (2004), 380--408.

[A-B3] Alexeev, V., Brion, M., Moduli of affine schemes with

reductive group action, J. Algebraic Geom., to appear.

[Ar] Arzhantsev, I. V. Actions of the group ${rm SL}sb 2$ that are of

complexity one. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997),

no. 4, 3--18; translation in Izv. Math. 61 (1997), no. 4, 685--698.

[B-Z] Berenstein, A., Zelevinsky, A., Tensor product multiplicities,

canonical bases and totally positive varieties. Invent. Math. 143(2001),

no.1, 77-128.

[Br1] Brion, M., Luna, D., Vust, Th., Espaces homogenes

spheriques. (French) [Spherical homogeneous spaces] Invent. Math. 84

(1986), no. 3, 617--632.

[Br2] Brion, M., Groupe de Picard et nombres caracteristiques des varietes

spheriques. (French) [Picard group and characteristic numbers of spherical

varieties] Duke Math. J. 58 (1989), no. 2, 397--424.

[Br3] Brion, M., The behaviour at infinity of the Bruhat

decomposition. Comment. Math. Helv. 73 (1998), no. 1, 137--174.

[Br4] Brion M., On orbit closures of spherical subgroups in flag

varieties, Comment. Math Helv. 76 (2001), 263-299.

[Co] Cox, David A. The homogeneous coordinate ring of a toric

variety. J. Algebraic Geom. 4 (1995), no. 1, 17--50.

[D-P] De Concini, C., Procesi, C. Complete symmetric varieties. Invariant

theory (Montecatini, 1982), 1--44, Lecture Notes in Math., 996, Springer,

Berlin, 1983.

[Dip] Dipper, Richard, James, Gordon The $q$-Schur algebra. Proc. London

Math. Soc. (3) 59 (1989), no. 1, 23--50.

[Don1] Donkin, S., On Schur algebras and related algebras. I. J.

Algebra 104 (1986), no. 2, 310--328.

[Don2] Donkin, Stephen, On Schur algebras and related

algebras. IV. The blocks of the Schur algebras. J. Algebra 168

(1994), no. 2, 400--429.

[Don3] Donkin, Stephen, The blocks of a semisimple algebraic group.

J. Algebra 67 (1980), no. 1, 36--53.

[Do1] Doty, Stephen, Polynomial representations, algebraic monoids,

and Schur algebras of classical type. J. Pure Appl. Algebra 123

(1998), no. 1-3, 165--199.

[Do2] Doty, Stephen, Representation theory of reductive normal algebraic

monoids. Trans. Amer. Math. Soc. 351 (1999), no. 6, 2539--2551.

[Ge] Geck, Meinolf, Modular Harish-Chandra series, Hecke algebras and

(generalized) $q$-Schur algebras. Modular representation theory of finite

groups (Charlottesville, VA, 1998), 1-66, de Gruyter, Berlin, 2001.

[Hal] Halverson, Tom, Representations of the $q$-rook monoid.

J. Algebra 273 (2004), no. 1, 227--251.

[Hal-R] Halverson, Tom; Ram, Arun, $q$-rook monoid algebras, Hecke

algebras, and Schur-Weyl duality, Zap. Nauchn. Sem. S.-Peterburg.

Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), Teor. Predst. Din.

Sist. Komb. i Algoritm. Metody. 6, 224--250, 262--263.

[H-S] Haiman, M., Sturmfels, B., Multigraded Hilbert schemes,

J. Algebraic Geom. 13 (2004), 725--769.

[H-T.a] Hassett, Brendan, Tschinkel, Yuri Universal torsors and Cox rings. Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 149-173, Progr. Math., 226, Birkhuser Boston, Boston, MA, 2004.

[H-T.b] Hassett, Brendan; Tschinkel, Yuri Geometry of equivariant

compactifications of $Gsb asp n$. Internat. Math. Res. Notices 1999,

no. 22, 1211-1230.

[Ka] Kato, S., A Borel-Weil-Bott type theorem for group

completions. J. Algebra 259 (2003), no. 2, 572-580.

[Ka] Kato, S., Equivariant vector bundles on group compactifications,

J. Crelle, to appear.

[K-K-M-S] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat,

``Toroidal embeddings I", Springer Verlag, New York, 1973.

[Kn] Knop, Friedrich The Luna-Vust theory of spherical

embeddings. Proceedings of the Hyderabad Conference on Algebraic Groups

(Hyderabad, 1989), 225-249, Manoj Prakashan, Madras, 1991.

[Lit] Littelmann, P., Bases canoniques et applications. Séminaire Bourbaki.

Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 847, 5, 287-306.

[L-V] Luna, D.; Vust, Th. Plongements d'espaces

homognes, Comment. Math. Helv. 58 (1983), no. 2, 186-245.

[Mok] Mokler, Claus The $mathbb F$-valued points of the algebra of strongly

regular functions of a Kac-Moody group. Transform. Groups 7 (2002),

no. 4, 343--378.

[P-S] Peeva, I., Stillman, M., Toric Hilbert schems, Duke Math J.

111(2002), 757-779.

[Pu] Putcha, Mohan S. Complex representations of finite

monoids. II. Highest weight categories and quivers. J. Algebra 205

(1998), no. 1, 53--76.

[P-R] Putcha, M. S., Renner, L. E. The system of idempotents and the

lattice of J-classes of reductive algebraic monoids. J. Algebra

116 (1988), no. 2, 385-399.

[Re1] Renner, Lex E. Analogue of the Bruhat decomposition for algebraic

monoids. J. Algebra 101 (1986), no. 2, 303-338.

[Re2] Renner, Lex Modular representations of finite monoids of Lie

type. J. Pure Appl. Algebra 138 (1999), no. 3, 279-296.

[Re3] Renner, Lex E. An explicit cell decomposition of the wonderful

compactification of a semisimple algebraic group. Canad. Math. Bull. 46

(2003), no. 1, 140-148.

[Rit] Rittatore, A. Algebraic monoids and group embeddings. Transform.

Groups 3(1998), no. 4, 375-396.

[Sol] Solomon, Louis An introduction to reductive monoids. Semigroups,

formal

languages and groups (York, 1993), 295-352, NATO Adv. Sci. Inst. Ser. C

Math. Phys. Sci., 466, Kluwer Acad. Publ., Dordrecht, 1995.

[Sp] Springer, T. A., Intersection cohomology of $Btimes B$-orbit

closures

in group compactifications. With an appendix by W. van der Kallen.

J. Algebra 258 (2002), no. 1, 71-111.

[Tc] Tchoudjem, A., Cohomologie des fibr'es en droites sur les

compactifications des groupes r'eductifs, Ann. Scient. Ecole

Norm. Sup. 37 (2004), 415--448.

[Ti1] Timashev, D. A., A generalization of the Bruhat

decomposition. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 58(1994),

no. 5, 110-123; translation in Russian Acad. Sci. Izv. Math. 45

(1995), no. 2, 339-352.

[Ti2] Timashev, D. A., Classification of $G$-manifolds of complexity

$1$. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 2,

127-162; translation in Izv. Math. 61 (1997), no. 2, 363-397.

[Ti3] Timashev, D. A., Cartier divisors and geometry of normal

$G$-varieties. Transform. Groups 5 (2000), no. 2, 181-204.

[Vi] Vinberg, E. B., On reductive algebraic semigroups. Lie groups and Lie

algebras: E. B. Dynkin's Seminar, 145-182, Amer. Math. Soc. Transl.

Ser. 2, 169, Amer. Math. Soc., Providence, RI, 1995.

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

numerical/cohomological information about representations of

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.

BIBLIOGRAPHY

[Al] Alexeev, V., Compactified Jacobians, ArXiv, 1996.

[A-B1] Alexeev, V., Brion, M., Stable reductive varieties I, Affine

varieties, Inventiones Math. 157 (2004), 227--274.

[A-B2] Alexeev, V., Brion, M., Stable reductive varieties II, Projective

varieties, Adv. Math. 184 (2004), 380--408.

[A-B3] Alexeev, V., Brion, M., Moduli of affine schemes with

reductive group action, J. Algebraic Geom., to appear.

[Ar] Arzhantsev, I. V. Actions of the group ${rm SL}sb 2$ that are of

complexity one. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997),

no. 4, 3--18; translation in Izv. Math. 61 (1997), no. 4, 685--698.

[B-Z] Berenstein, A., Zelevinsky, A., Tensor product multiplicities,

canonical bases and totally positive varieties. Invent. Math. 143(2001),

no.1, 77-128.

[Br1] Brion, M., Luna, D., Vust, Th., Espaces homogenes

spheriques. (French) [Spherical homogeneous spaces] Invent. Math. 84

(1986), no. 3, 617--632.

[Br2] Brion, M., Groupe de Picard et nombres caracteristiques des varietes

spheriques. (French) [Picard group and characteristic numbers of spherical

varieties] Duke Math. J. 58 (1989), no. 2, 397--424.

[Br3] Brion, M., The behaviour at infinity of the Bruhat

decomposition. Comment. Math. Helv. 73 (1998), no. 1, 137--174.

[Br4] Brion M., On orbit closures of spherical subgroups in flag

varieties, Comment. Math Helv. 76 (2001), 263-299.

[Co] Cox, David A. The homogeneous coordinate ring of a toric

variety. J. Algebraic Geom. 4 (1995), no. 1, 17--50.

[D-P] De Concini, C., Procesi, C. Complete symmetric varieties. Invariant

theory (Montecatini, 1982), 1--44, Lecture Notes in Math., 996, Springer,

Berlin, 1983.

[Dip] Dipper, Richard, James, Gordon The $q$-Schur algebra. Proc. London

Math. Soc. (3) 59 (1989), no. 1, 23--50.

[Don1] Donkin, S., On Schur algebras and related algebras. I. J.

Algebra 104 (1986), no. 2, 310--328.

[Don2] Donkin, Stephen, On Schur algebras and related

algebras. IV. The blocks of the Schur algebras. J. Algebra 168

(1994), no. 2, 400--429.

[Don3] Donkin, Stephen, The blocks of a semisimple algebraic group.

J. Algebra 67 (1980), no. 1, 36--53.

[Do1] Doty, Stephen, Polynomial representations, algebraic monoids,

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