Инварианты сферических многообразий тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Авдеев Роман Сергеевич

• Авдеев Роман Сергеевич
• доктор наук
• 2025

Оглавление диссертации доктор наук Авдеев Роман Сергеевич

Contents

llntroductionl

H

9

Preliminaries!

11.1 Xotatiori and conventions!

11.1.1 (General notation!

.Notation for reductive groups and related objects!

1

1

Cones, fans, and Dcmazurc roots|

1.2 General facts from algebraic groups!

1.2.1 Levi roots and their properties!

1.2.2 Regular embedding of a subgroup!

1.2.3 Structure of K[G]

1.2.4 Intersection of kernels of all characters of an algebraic group! ■ ■ ■ ■

1.2.5 Reductive subgroups with characters of infinite order|

1.2.6 Structure constants of Chevallev bases!

1.3 General facts from algebraic geometry!

1.3.1 A general property of smooth morphisms|

1.3.2 One-parameter degenerations in complete varieties!

1.3.3 Additive degenerations of subspaces in simple si2-modules

1.3.4 Homogeneous bundles]

1.3.5 Linearized line bundles on homogeneous spaces!

1.4 Generalities on root subgroups!

1.4.1 Ga-actions and locally nilpotent derivations

1.4.2 Equivalence of Ga-subgroups

1.4.3 Root subgroups!

1.4.4 Orbits of root subgroups!

1.4.5 Root subgroups on torie varieties!

2 Generalities on spherical varieties

2.1 The principal combinatorial invariant?

2.1.1 Stabilizer of the open 5-orbit

2.1.2 Weight lattice

2.1.3 Spherical roots

48

2.1.5 Uniqueness property for spherical homogeneous spaces!

2.1.6 Typos of simple roots and colors|

12.1.7 Modified notation for invariants!

O The extended weight monoid|

12.2.1 Definite

2.2.2 Computing the principal combinatorial invariants via the extended

weight monolcfl

2.2.3 Recovering the extended weight monoid from the principal combinatorial invariants!

2.2.4 Affinc spherical homogeneous spaces and their extended weight

monoids!

2.3 Simple embeddings of spherical homogeneous spaces|

2.3.1 Definition and classification of simple embeddings

2.3.2 Standard completions, wonderful G-varieties, and Demazure em-

beddings

2.3.3 Affine embeddings of spherical homogeneous spaces

2.4 Multiplicity-free varieties and affine spherical varietie s

2.4.1 Multiplicity-free quasi-affine G-varieties

2.4.2 The weight monoid

12.4.3 The root monoid and related invariants!

2.4.4 Horospherical varieties

2.4.5 Degenerations of multiplicity-free affine G-varieties

2.4.6 Further properties of affine spherical varieties

2.5 Spherical modules

2.5.1 Basic properties

2.5.2 A reduction for spherical modules

2.5.3 Classification of spherical modules and some consequences

2.6 Spherical subgroups

2.6.1 Sphericity criterion

2.6.2 The weight lattice of a spherical homogeneous space

12.6.3 Parabolic induction!

2.6.4 -Normalizers of spherical subgroups

2.6.5 The spherical closure of a spherical subgroup

2.7 Luna's general classification of spherical homogeneous spacc s

2.7.1 Classification of spherical homogeneous spaces and wonderful G-

varieties!

2.7.2 Distinguished subsets of colors and quotient systems

2.7.3 Characterization of strongly solvable spherical subgroups

2.8 Luna's 1993 approach for classifying strongly solvable wonderful subgroups|

2.8.1 Spherical and wonderful B-

varieties

91

2.8.2 Smooth complete spherical B -varieties and Enriques' B -systems

2.8.3 Wonderful B -varieties and admissible maps|

3

Explicit classification of strongly solvable spherical subgroups |

Sphericity criterion and some applications!

3

3

3

372 Active root theorv

Sphericity criterion!

First applications and a uniqueness result!

3.2.1 Active roots, their properties, and classification!

3.2.2 Intersection of supports of two active roots|

3.2.3 The support of a maximal active root|

3.2.4 Relation between S and ^

ARS-sets, extended ARS-sets, and uniqueness theorem!

3

13.3 Kxisterice theoreml

13.3.1 Statement of the theoreml.......................Ill

3.3.2 Properties of reconstructed active roots]................Ill

3.3.3 Construction of the Lie algebra of the unipotcnt radical!

EI

Construction of the strongly solvable spherical subgroup!

118

3.4 Classification of strongly solvable spherical subgroups up to conjugation

3.4.1 Preliminary results!

3.4.2 Elementary transformations and the main result!

3.4.3 Changes in the combinatorial data under elementary transformations|L20

3.4.4 Description of all regular active roots|

3.4.5 Classification of sober connected solvable spherical subgroups!

375 Simplification of the set of combinatorial data corresponding to a strongly

solvable spherical subgroup!

Reduction to a simplified form of extended ARS-scts|

3

3.5.2 Conjugacy conditions for the simplified form of extended ARS-sets],

3.5.3 Elementary transformations preserving the simplified form of extended A US-sot,si

376 Xormalizers of solvable spherical subgroups!

13.6.1 Statement of the results!

Further results on connected solvable spherical subgroups!

3

13.6.3 Proofs of the main results!

3.7 Examples!

4 Combinatorial invariants of strongly solvable spherical subgroups

4.1 Regular embedding in B

4.2 Extended weight monoids via the explicit classification

14.2.1 Statement, of the result]

4.2.2 The rank of the extended weight monoid

4.2.3 Candidates for indecomposable elements

4.2.4 Indecomposabilitv

14.3 Interrelations between the three classifications!

4.3.1 The spherical closure of a strongly solvable spherical subgroup

4.3.2 Relationship between Luna's 1993 classification and Luna's general

classification!

4.3.3 Principal combinatorial invariants via the explicit classification!

4.3.4 Relationship between the explicit classification and Luna's

classification!

ftt

4.3.5 Computation of extended ARS-scts via homogeneous spherical data] 164 Strongly solvable wonderful subgroups in small rank|

5 Degenerations of spherical subalgebras and spherical roots

5.1 The general strategy for computing the set of spherical roots

5.2 Active G-roots and their properties

5.3 Implementation of the general strategy

171

15.3.1 Outline

5.3.2 Description of the group NG(H)0

5.3.3 Reduction of the ambient group!

5.3.4 Multiplicative degenerations

5.3.5 Additive degenerations

5.4 An optimization of the base algorithm for the case K = L

15.4.1 The main ideal

5.4.2 Auxiliary results!

5.4.3 The subgroup Hi

15.4.4 The main resTiTtl

5.4.5 A further optimization

5.5 Examples

5.6 Concluding remarks

6 Extended weight monoids of arbitrary spherical homogeneous spaces

6.1 Statement of the results and applications

203

16.1.1 Statement of the main results!

6.1.2 The strongly solvable case|

6.1.3 Examples!

16.2 Proofs of the main result,si

6.2.1 Préparations]

16.2.2 Proofs!

6.3 Concluding remarks!

7 Moduli schemes of affine spherical varieties with a given weight monoid g

7.1 Moduli schemes Mr

7.1.1 The definition of Mr

7.1.2 Relation of Mr to multiplicity-free affine G-varieties with weight

monoid r

7.1.3 Basic facts on the action of Tad on Mr

7.1.4 Characterizations of TXoMr

7.2 The tangent space of Mr at X0

17.2.1 Statement, of the main result,!

7.2.2 Preliminaries for the proof of Theorem [7.2.2|

7.2.3 The role of saturatedness of r

7.2.4 Basic properties of {V/qx0)gx° and it s Tad-weights

7

17.2.5 Extension of sectionsi

Canonical representatives of Tad-eigenvectors in TS

7.2.7 Proof of Theorem |7.2.2| StepT

7.2.8 Proof of Theorem |7.2.2[ Step

7.3 Applications to affinc spherical and multiplicity-free varieties|

7.3.1 Auxiliary results on $(r)

7.3.2 Applications to affine spherical G-varieties

7.3.3 Consequences for multiplicity-free affine G-varieties

7.3.4 .New proof of the uniqueness property for spherical homogeneous"

spaces

256

7.4 Affine spherical G-varieties with a prescribed weight monoid

7.4.1 Spherical roots compatible with a lattice]

7.4.2 Spherical roots compatible with a monoid|

7.4.3 Admissible sets of spherical roots for a given monoid|

7.5 Applications to the geometry of moduli schemes Mr

7.5.1 Additional properties of Mr

7.5.2 A combinatorial description of the irreducible components of Mr , ,

7.5.3 The tangent space of Mr at X0 revisited

7.5.4 A smoothness criterion for Mr

7.5.5 Sufficient conditions for Mr to be irreducible and/or smooth

7.5.6 Examples of reducible Mr

7.5.7 Examples where Mr is a non-reduced point

8 Root subgroups on affine spherical variétés

8.1 Basic properties of 5-root subgroups

8.1.1 First properties of 5-root subgroups

8.1.2 Vertical and horizontal 5-root subgroups

8.1.3 Simplest constructions of 5-root subgroups and/or 5-normalized

1 lADsl

8.1.4 Multiple 5-root subgroups of the same weight

8.1.5 Weights of 5-root subgroups

8.1.6 5-stable prime divisors moved bv 5-root subgroups

8.1.7 The case of G-saturated r

8.2 Standard B-root subgroups

8.2.1 Description of the construction

8.2.2 G-stable prime divisors moved by standard 5-root subgroups

8.2.3 G-root subgroups

8.2.4 The horospherical case

8.3 Local description of 5-root subgroups

284

18.3.1 Local structure theoreml

8.3.2 5-root subgroups on an open cell

8,3,3 Extension of 5-root subgroups from an open cell IConcliisionl

|The author's publications on the topic of the thesis Bibliography

Appendix A: Russian translation of the thesis

290

294

295

Введение диссертации (часть автореферата) на тему «Инварианты сферических многообразий»

Introduction

Throughout this thesis we work over an algebraically closed field K of characteristic 0,

Relevance of the topic and its development degree

Spherical varieties, their principal combinatorial invariants, and the classification problem. Let G be a connected reductive algebraic group and let B be a Borel subgroup of G, A G-varietv (that is, an algebraic variety equipped with a regular action of G) is said to be spherical if it is irreducible, normal, and possesses an open 5-orbit, In particular, one may speak of spherical homogeneous spaces and spherical (finite-dimensional) G-modules, A dosed subgroup H C G is said to be spherical if G/H is a spherical homogeneous space. The Lie algebras of spherical subgroups in G are said to be spherical subalgebras of the Lie algebra g = Lie G. The sphericity property admits various equivalent characterizations; an extensive list of them, as well as proofs of the

equivalences, can be found in the monograph by Timashev; see |Timll §25|.

Spherical varieties form an extremely interesting class of G-varieties. The most famous representatives of this class are toric varieties, flag varieties, and symmetric varieties. Countless papers are devoted to various aspects concerned with spherical varieties or

certain subclasses of them. Some of these aspects are reflected in |Timll Chapter 5|, In

this context we also mention the survey paper |Perl4 devoted to the geometry of spherical varieties.

For an arbitrary homogeneous space G/H^ every normal irreducible G-varietv containing G/H as an open orbit is said to be an equivariant embedding (or simply an embedding) of G/H. In this terminology, spherical G-varieties are precisely embeddings of spherical homogeneous spaces.

An important problem in the theory of spherical varieties is that of classifying them. The spherical G-varieties with a given open G-orbit, that is, the embeddings of a given spherical homogeneous space, were classified by Luna and Vust in the framework of their general theory of embeddings of arbitrary homogeneous spaces developed in | LV83|, Later, in the particular case of spherical varieties this theory was considerably simplified and

restated in a more transparent form by Knop |Kno91 , As a result, the embeddings of a given spherical homogeneous space are classified in terms of certain objects of convex geometry called colored fans, which generalize usual fans used for classifying toric varieties. Thus the classification of spherical varieties reduces to that of spherical homogeneous spaces, which by definition is equivalent to the classification (up to eonjugaey) of spherical

subgroups in reductive algebraic groups.

In classifying spherical homogeneous spaces G/H (or, equivalentlv, spherical subgroups H C G up to conjugacv) one may restrict the consideration to the case where G is semisimple; see, for instance, [VinOl, §1.3,3, Corollary 2] or [Timll, §10,2], For semisimple G, a complete classification of affine spherical homogeneous spaces G/H (that is, with reductive #) was obtained by Kramer [Kra79| (in the case of simple G), Mikitvuk [Mik87|, Brion [Bri87| (both treated independently the case of nonsimple semisimple G), and Yaki-mova |Yak02| (small corrections). We note that this classification is essentially given by a list of spaces and does not involve any combinatorial invariants.

At present, there is a complete classification in combinatorial terms of spherical homogeneous spaces G/H, which is a result of joint efforts of several researchers. The idea of this classification was proposed by Luna in 2001 |Lun01 , therefore we shall refer to

it as Luna's general classification. This classification is carried out in two steps. The first one is to reduce the classification to the case of so-called wonderful subgroups, A spherical subgroup H C G is said to be wonderful if G/H admits a wonderful completion, which is a smooth complete embedding with certain additional properties (see Defini-

tions 2,3,11 and 2,3,12), Wonderful completions of spherical homogeneous spaces G/H are also known as wonderful G-varieties, They are generalizations of complete symmetric varieties considered in |DP83|, The second step of the classification is to describe all the wonderful G-varieties, In the paper [Lun01|, Luna himself performed the first step in full generality, stated a conjecture for the second step, and proved this conjecture in the case where G is a product of simple groups of type A, Luna's conjecture claimed that wonderful G-varieties are classified by combinatorial objects called spherical systems. During the following several years the conjecture was proved in certain other particular cases; BP05|, |Bra07|, and |BC10 , The uniqueness part of Luna's conjecture was proved by

see

Losev |Los09a| in 2009 by a general argument, A complete proof of the existence part,

which follows the lines of the original constructive approach employed by Luna, was ob-

tained by Bravi and Pezzini in the series of papers |BP14 , |BP15 , and |BP16 (see also an

alternative approach via moduli schemes in |Cupl4 ), We also mention the paper | BL11

which is an introduction to wonderful varieties with an emphasis put on the combinatorics of spherical systems,

Luna's general classification involves four combinatorial invariants n = nX, A = Ax, £ = £x, V = Vx of a spherical G-variety X, where

• n is the subset of simple roots of G encoding the stabilizer of the open 5-orbit in X (this is a parabolic subgroup containing B);

• A is the weight lattice, defined as the lattice of weights of 5-semi-invariant rational functions on X;

• £ is the set of spherical roots of X, which is a linearly independent set of primitive elements in A controlling the set of G-invariant discrete valuations of the field of rational functions on X;

• V is the (finite) set of colors of X, which are 5-stable but not G-stable prime divisors in X; V comes equipped with a map k : V ^ HomZ(A, Z),

In this thesis, the invariants W, A, E, V are referred to as the principal combinatorial invariants. They come from the Luna-Vust theory mentioned above: the description of all embeddings of a fixed spherical homogeneous space is given in terms of the triple (A, E, V). A detailed discussion of the principal combinatorial invariants, including precise definitions and basic properties, can be found in § ^TT]

We note that the above-described results provide a complete combinatorial description of all spherical G-varieties X, which is given by data encoding the open G-orbit O C X and data enconding the embedding of O in X,

A subgroup H C G is said to be strongly solvable if it is contained in a Borel subgroup of G. We note that every connected solvable subgroup of G is automatically strongly solvable. An example of a solvable but not strongly solvable subgroup is given by the normalizer of a maximal torus in SL2,

Apart from Luna's general classification of arbitrary spherical subgroups, there is an additional combinatorial classification in the case of strongly solvable spherical subgroups.

It was obtained by Luna in his unpublished 1993 preprint |Lun93 ; see also a detailed

exposition in |D5, §4|, This classification applies only to the case of wonderful strongly solvable subgroups. In what follows we shall refer to this classification as Luna's 1993 classification.

Both Luna's general classification and Luna's 1993 classification have a geometric origin: the invariants involved arise from the geometry of the corresponding homogeneous spaces. As a result, both classifications provide no simple method for relating the invariants with an explicit description of the corresponding subgroups. In this sense both Luna's classifications are "implicit", A standard way of explicitly specifying a subgroup H C G is to use a regular embedding of H in a parabolic subgroup P C G, where "regular" means the inclusion Hu C Pu of the unipotent radicals of H and P, respectively. Thus, the following two problems are very natural:

(PI) compute the principal combinatorial invariants of a spherical subgroup specified by a regular embedding in a parabolic subgroup of G;

(P2) determine (in the same sense) the subgroup corresponding to a given set of principal combinatorial invariants.

As was already mentioned above, there is a complete classification of all affine spherical

homogeneous spaces G/H (that is, with reductive H) due to [Kra79, Mik87, Bri87 , A description of the sets np, weight lattices, and colors for this case follows from results Kra79,Dl|, and the sets of spherical roots are known thanks to the paper |BP15

m

Thus problem (|Pl]) is solved in this case. Since the classification is essentially given by a

list of spaces, problem (|P2[) is also solved automatically.

For arbitrary spherical subgroups, the state of the art in solving problem (PI) is as

follows. First, there is a general method tracing back to Panyushev |Pan94 for computing the weight lattice of a spherical homogeneous space G/H in terms of a regular embedding of H in a parabolic subgroup of G. As a byproduct, one also obtains a description of the set np. Second, the author is not aware of any general results on computing the colors. Third, there are two different general approaches for computing the set of spherical roots: one is

due to Luna and Vust (the method of formal curves; see |LV83: §4| or |Timll. §24|) and the other one is due to Losev |LoslO|, (In fact, both approaches deal with a generalization of the set of spherical roots to arbitrary G-varieties.) However, these two approaches seem to work effectively only for some special classes of spherical homogeneous spaces. As a consequence, the problem of finding effective algorithms for computing the spherical roots and colors for arbitrary spherical subgroups still remains of great importance.

As for problem (|P2j), a considerable progress in solving it for wonderful subgroups was

3,51 and |BP16 , Namely, their approach in

achieved bv Bravi and Pezzini in BP14

fact enables one to construct a wonderful subgroup H starting from its spherical system; however, the procedure is very indirect and consists in several reduction steps leading in the end to a list of "primitive" cases. From this procedure, one can extract an explicit description of a Levi subgroup of H. As for determining the unipotent radical Hu C H, they suggested a technique that helps to guess Hu in every concrete example. Unfortunately, so far there is no proof that this technique will always work,

A major part of this thesis is devoted to solving problems (|PT) and/or (|P2} under various restrictions.

Weight monoids and extended weight monoids. Let X be a normal irreducible G-variety. According to a result of Vinberg and Kimelfeld [VK78 Theorem 2], X being

spherical implies that the algebra K[X] of regular functions on X is multiplicity free as a G-module, and the converse is also true if X is quasi-affme. The highest weights of all simple G-modules that occur in K[X] form a monoid, called the weight monoid (or weight ■semigroup) of X; we denote it by rx. If X is spherical, then rx uniquely determines the G-module structure on K[X] and hence is an important invariant of X,

Now suppose X = G/H is a homogeneous space. Then another result of Vinberg

and Kimelfeld [VK78, Theorem 1] asserts that G/H being spherical is equivalent to the following property: for every G-linearized line bundle £ on G/H, the space r(£) of global sections of £ is multiplicity free as a G-module, Thanks to the Frobenius reciprocity theorem (see |VK78. §2| or |Timll. Corollary 2,13|), the latter property can be reformulated as follows: for every simple finite-dimensional G-module V and every character x of H, the subspace V^CH^ of #-semi-invariant vectors in V of weight x ^ at most one-dimensional

(see details in § 2,2,1),

Let X(B),X(H) be the character groups of B,H (in additive notation), respectively, and let AJ C X(B) be the monoid of dominant weights of G with respect to B. For every A e AG, denote bv VG(A) the simple G-module with highest weight A and let VG(X)* be the respective dual G-module, Then the set

r

G/H

|(A,X) e AG x X(H) | [Vg(X)*][h)=0> is a monoid, called the extended weight monoid (or extended weight semigroup) of G/H.

The term "extended weight monoid" was introduced in the paper |D1 though the monoid itself appeared implicitly many times in earlier papers of different authors. The word "extended" is justified by the fact that the map A m (A, 0) identifies r G/ H with the submonoid |(A,x) | X = 0} C TG/ H. In particular, rG/H ~ rG/H whenever X(H) is trivial. If G/H is spherical, then rG/H uniquely determines the G-module structures

on spaces of global sections of all G-linearized line bundles on G/H, therefore a natural problem is to find explicit formulas and/or algorithms for computing the extended weight monoids for spherical homogeneous spaces.

The extended weight monoid of a spherical homogeneous space is closely related to the principal combinatorial invariants, Namely, it turns out that, knowing the extended weight monoid, one can compute all the principal combinatorial invariants except for the spherical roots, and in the strongly solvable case this monoid determines all the principal combinatorial invariants. On the other hand, the extended weight monoid is recovered from the principal combinatorial invariants, A systematic study of the interrelations between these invariants is undertaken in |D5. §2,3| and reproduced in §§ |2.2.2| p.2.3|for

convenience.

As was mentioned above, to solve problem (|PTj) in the general case it suffices to find methods and/or algorithms for computing the spherical roots and the colors. As follows from the previous paragraph, computing the colors may be replaced with computing the extended weight monoid. In fact, there is a close relation between the extended weight monoid and the colors; see the next paragraph.

For a general spherical homogeneous space G/H, computing the monoid Tg/h easily reduces to the case where G is semisimple and simply connected (see Remark 2,2,4), In

the latter case it is well known that TG/H is naturally isomorphic to the monoid of B-stable effective divisors on G/H (see Theorem 2.2.2); in particular, TG/H is free and its indecomposable elements are in bijection with the set VG/H of colors. More specifically, given a color D e VG/h, the corresponding indecomposable element of rG/# is precisely the (B x H)-biweight of a (unique up to proportionality) regular function on G whose divisor of zeros is the pullback of D via the map G ^ G/H.

Recall that there is a complete classification of all affine spherical homogeneous spaces G/H (that is, with reductive H) due to [Kra79, Mik87, Bri87|, For all such spaces with G semisimple and simply connected and H connected, the monoids rG/# are known; their description either was obtained in or easily follows from results of |Kra79. D1

We

reproduce these results in §|2.2.4

An interesting application of the extended weight monoids of spherical homogeneous spaces to matrix-valued orthogonal polynomials can be found in | PVP23 ,

Moduli schemes of multiplicity-free affine G-varieties with a prescribed weight monoid. An affine G-varietv X is said to be multiplicity-free if X is irreducible and the algebra K[X] of regular functions on X, regarded as a G-module, contains every simple G-module with multiplicity at most 1, By a theorem of Vinberg and Kimelfeld [VK78, Theorem 2], an irreducible affine G-varietv is multiplicity-free if and only if it possesses a dense B-orbit. In particular, affine spherical G-varieties are characterized as normal multiplicity-free affine G-varieties,

Given a multiplicity-free affine G-varietv X, the G-module structure of K[X] is encoded in the weight monoid rx of X, consisting of all dominant weights A of G for which K[X] contains a simple G-submodule K[X]A with highest weight A, This monoid is known to be finitely generated. Besides, X is normal if and onlv if rx is saturated, that is, Tx is the intersection of a lattice with a cone. Recall that weight monoids for spherical G-varieties

have already been discussed above.

One more invariant of a multiplicity-free affine G-variety X te its root monoid Ex, which arises from the ring structure of K[X]. By definition, is generated by all expressions À + y — v with G rx such that the linear span of K[X]A ■ K[X

contains K[X]^, Let denote the saturation of Ex, that is, the intersection of the lattice generated by Ex with the cone spanned by Ex- An important property of the root

:sat jg free_

monoid was discovered by Knop in |Kno96|, who proved that the monoid

In |AB05|, Alexeev and Brion constructed and studied a moduli scheme Mr for mul-tiplicitv-free affine G-varieties with prescribed weight monoid T, This scheme is affine and of finite type; it is equipped with an action of an adjoint torus Tad (the quotient of a maximal torus of G by the center of G) in such a way that the Tad-orbits of Mr bijectivelv correspond to the G-equivariant isomorphism classes of multiplicity-free affine G-varieties with weight monoid T, Various examples of moduli schemes Mr were further studied under different assumptions on the monoid T, The case of monoids generated by

a single element was worked out in |Jan07 ; the paper |BC08 dealt with free monoids that are G-saturated (the latter means that the monoid consists of all dominant weights of G lying in a fixed lattice); several other special instances of free monoids were studied PVS12,PVS16|, In all these cases, Mr was shown to be an affine space (as a scheme).

m

Finally, in [BVS16| it was proved that for an arbitrary free monoid r all the irreducible components of Mr, equipped with their reduced subscheme structure, are affine spaces.

Given an arbitrary finitely generated monoid r of dominant weights of G, there always exists a multiplicity-free affine G-variety X0 = X0(r) with weight monoid r such that the linear span of K[X0]A ■ K[X0]M coincides with K[X0]A+M to all A,^ e T, Such varieties were first considered and studied by Vinberg and Popov in [VP72 under the name ' S-

varieties'; in the case of saturated r they are also known as affine horospherical (spherical) G-varieties; see § 2,4,4 for details. As shown by Popov in [Pop86 , the G-varietv X0 is a

common G-equivariant degeneration of all multiplicity-free affine G-varieties with weight monoid r. It is known from [AB05| that the Tad-orbit in Mr corresponding to X0 is just a Tad-fixed closed point (still denoted by X0), hence the tangent space TXoMr of Mr at X0 is naturally equipped with the structure of a Tad-module,

In this thesis we obtain a complete description of the Tad-module structure in TXo Mr and apply it to the study of various properties of affine spherical G-varieties as well as the geometry of the moduli scheme Mr,

Root subgroups. Let Ga := (K, +) be the additive group of the field K regarded as an algebraic group. If G„ acts nontriviallv on an irreducible algebraic variety X, its image H in the automorphism group Aut(X) is called a Moreover, if X is

equipped with a regular action of a linear algebraic group F and H is normalized by F, then we call H an F-root subgroup on X. In this case, the action of F on H by conjugation is controlled by a character of F, called the weight of H.

When F = T is a torus, T-root subgroups on T-varieties are used to study the automorphism groups

AHHL14 . AL12 LielOa LielOb Xil06

tomorphism groups |AKZ12 AKZ19 , equivariant group embeddings |AK15 AR17

transitivity properties for au-

and

affine algebraic monoids | ABZ20, DZ21 Bil22 Zai24 , The simplest in this setting is the

classical case of toric varieties, that is, normal irreducible T-varieties containing an open T-orbit, Toric varieties admit a complete combinatorial description in terms of objects of convex geometry called fans [CLS11, Ful93, Oda88 , and the T-root subgroups on a given toric T-varietv are described in the following simple way: every such subgroup is uniquely determined by its weight and the set of all possible weights is the collection of so-called Demazure roots of the associated fan ipem70|Dda88||Cox95| ILielOal; see also [IA.Z24 ,

A natural intention is to extend the above picture to algebraic varieties equipped with an action of an arbitrary connected reductive algebraic group G. In this setting, a proper generalization of toric varieties is given by spherical varieties. The problem of describing all G-root subgroups on affine spherical G-varieties was raised in the preprint [LP 14 , It

is shown there that such a subgroup is uniquely determined by its weight, and the set of weights is described in some particular cases. However, the set of G-root subgroups on an affine spherical G-varietv seems to be quite restricted; in particular, it is empty whenever G is semisimple.

In this thesis, we study B-root subgroups on affine spherical G-varieties, where B is a Borel subgroup of G. Unlike G-root subgroups, we believe that 5-root subgroups are more natural and suitable generalizations of T-root subgroups on T-varieties,

By now, B-root subgroups on affine spherical G-varieties have already appeared in the literature: in [RVS21|, 5-root subgroups are used to prove that a smooth affine spherical

G-varietv not isomorphic to a torus is uniquely determined by its automorphism group in the category of smooth affine irreducible varieties. In our thesis, 5-root subgroups on affine spherical G-varieties naturally arise and play an important role in the study of extended weight monoids of spherical subgroups; see §^3]for details.

Results of this thesis on B-root subgroups on affine spherical G-varieties are discussed below.

The study of B-root subgroups on (not necessarily affine) spherical G-varieties seems to be an interesting and promising research area; further developments in this direction bevond this thesis can be found in IAZ24

Objectives of the thesis

(1) To develop a structure theory of strongly solvable spherical subgroups in connected reductive algebraic groups and obtain an explicit classification of all such subgroups up to conjugation,

(2) To compute the normalizers of all connected solvable spherical subgroups in connected reductive algebraic groups and thereby complete a classification of all (not necessarily connected or strongly) solvable spherical subgroups in connected reductive algebraic groups,

(3) For every strongly solvable spherical subgroup H in a simply connected semisimple algebraic group G, to compute the extended weight monoid of G/H in terms of the structure theory mentioned in (|l|.

(4) To establish relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in connected reductive algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the explicit classification mentioned in ([!}.

(5) To obtain several structure results for a class of spherical subgroups of connected reductive algebraic groups that extends the class of strongly solvable spherical subgroups, Based on these results, to construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups and apply them for computing the sets of spherical roots of such subgroups,

(6) For an arbitrary spherical subgroup H of a connected reductive algebraic group G such that H is specified by a regular embedding in a parabolic subgroup P C G, to obtain a description of the extended weight monoid of G/H via the set of simple spherical roots of G/H together with certain combinatorial data explicitly computed from the pair (P, H),

(7) For every finitely generated and saturated monoid r of dominant weights of a connected reductive group G, to obtain a complete description of the Tad-module structure in the tangent space of the moduli scheme Mr at the distinguished Tad-fixed point X0. To apply this description for studying root monoids of affine spherical G-varieties, for obtaining new proofs of several uniqueness results on spherical varieties, and for exhibiting reduced subschema structures on the irreducible components of Mr.

(8) To obtain a combinatorial description of all (G-equivariant isomorphism classes of) affine spherical varieties with prescribed weight monoid T, As an application, to obtain a characterization of the irreducible components of the moduli scheme Mr for such varieties,

(9) To propose a construction of standard B-root subgroups on affine spherical G-varieties, which generalizes the well-known construction of T-root subgroups on affine toric T-varieties, To apply this construction for studying various properties for B- and G-root subgroups on affine horospherical G-varieties,

(10) To obtain a complete description of all B-root subgroups on an open B-stable subset of an arbitrary affine spherical G-variety X, Using this description, to obtain several sufficient conditions for extending B-root subgroups from the open subset to the whole X.

Theoretical and practical significance of the thesis

The thesis is theoretical in nature. Its results and methods can be used in further research on the theory of algebraic groups, algebraic transformation groups, Lie algebras, algebraic geometry, and representation theory.

Main research methods

In the thesis we use methods from algebraic geometry, the theory of algebraic transformation groups, representation theory, as well as the structure theory of algebraic groups and Lie algebras.

Main results of the thesis

Explicit classification of strongly solvable spherical subgroups. The first contribution of this thesis is a new classification of strongly solvable spherical subgroups in connected reductive algebraic groups; see Chapter [I] In contrast to both Luna's classifications mentioned above, our approach is much more algebraic: the invariants involved encode explicitly an embedding of the Lie algebra h = Lie H in g = Lie G, so that H can be easily recovered from the corresponding invariants. This is why we call our classification "explicit".

Let us briefly explain the main ideas of the explicit classification. Let H C B be an arbitrary subgroup, let U, N be the unipotent radicals of B, H, respectively, and fix a maximal torus T C B, so that B = T A U. One may always assume that S = T n H is a Levi subgroup of in which case we say that H is regularly embedded, in B. Let A+ be the set of positive roots of G with respect to T and B and to every a e A+ let ga be the corresponding root subspace, so that the Lie algebra u = Lie U decomposes as u =0 0a- If H is spherical, then the crucial object assigned to H is the set of

a€A+

active roots ^ = [a e A+ | ga C h}- We prove that, up to conjugation by an element of T, a spherical subgroup H C G regularly embedded in B is uniquely determined by the pair (S, After that, we perform an extensive study of various combinatorial properties of the set ^ and its relations with S, which leads to a structure theory of strongly solvable spherical subgroups and to a complete classification of all possible pairs (S, The latter gives a classification of all spherical subgroups H C B up to conjugation by elements of T. At the last step we determine when two pairs (Si, ^i) and (S2, ^2) correspond to spherical subgroups conjugate in G, which completes the classification of all strongly solvable spherical subgroups in G up to conjugation.

Заключение диссертации по теме «Другие cпециальности», Авдеев Роман Сергеевич

Заключение

В настоящей диссертации мы исследовали несколько вопросов, связанных с инвариантами сферических многообразий,

В главе |3] мы разработали структурную теорию сильно разрешимых сферических подгрупп в связных редуктивных алгебраических группах и применили её для получения явной классификации таких подгрупп в терминах некоторых корневых данных, кодирующих алгебру Ли соответствующей подгруппы. Более того, эта структурная теория позволила нам распространить классификацию на произвольные (не обязательно связные или сильно) разрешимые сферические подгруппы,

В главе[|мы изучили различные комбинаторные инварианты сильно разрешимых сферических подгрупп и установили взаимосвязи между нашей явной классификацией, общей классификацией Луны и классификацией Луны 1993 г, таких подгрупп. В частности, для каждой из двух классификаций мы нашли явные формулы или разработали эффективные алгоритмы, позволяющие выразить инварианты, участвующие в одной из этих классификаций, стартуя с инвариантов, участвующих в другой, В частности, наши результаты дают полные решения обеих задач (pl) и (р2) для сильно разрешимых сферических подгрупп.

В главах |5[ |б] мы сосредоточились на задаче (Р1_| для произвольных (не обязательно сильно разрешимых) сферических подгрупп связных редуктивных алгебраических групп. Как следует из теоремы [2.2.181 основные комбинаторные инварианты сферического однородного пространства однозначно определяются набором сферических корней и расширенным весовым моноидом, поэтому для решения задачи (Р1) достаточно вычислить только эти два инварианта.

В главе [5]мы предложили общую стратегию для вычисления набора сферических корней сферического однородного пространства G/H, которая сводится к построению вырождений алгебры Ли (), принадлежащих G-орбитам коразмерности 1 во вложении Демазюра пространства G/H. Затем мы реализовали общую стратегию для класса сферических подгрупп, включающего в себя сильно разрешимые подгруппы. А именно, рассматриваемый класс состоит из сферических подгрупп H, регулярно вложенных в некоторую параболическую подгруппу Р С G с подгруппой Леви L таким образом, что подгруппа Леви группы H лежит между L и её коммутантом. Несколько направлений дальнейших исследований, связанных с методами и результатами главы ¡5]. обсуждаются в § |Гб

В главе |б] мы разработали метод вычисления расширенного весового моноида произвольного сферического однородного пространства G/H по модулю набора простых сферических корней пространства G/H. В результате решение задачи (Р1) в полной

общности сводится к нахождению методов вычисления множества простых сферических корней дня произвольных сферических подгрупп. Некоторые идеи в этом направлении обсуждаются в §|6.3[

В главе [7] мы изучили несколько аспектов, связанных с пространствами модулей Мг Алексеева и Бриона для аффинных сферических С-многообразий с заданным весовым моноидом Г. Мы нашли полное описание касательного пространства к Мг в наиболее вырожденной точке Х0.; которая соответствует аффинному орисфериче-

СГ что корневой моноид любого аффинного сферического многообразия свободен, получили новые доказательства результатов Лосева о единственности дня аффинных сферических многообразий и сферических однородных пространств, получили ноС

Мг

жённые приведённой структурой подсхемы, являются аффинными пространствами. Используя известную классификацию сферических однородных пространств и их вложений, мы вывели комбинаторное описание всех аффинных сферических многообразий с заданным весовым моноидом. Мы применили это описание дня харак-

Мг

Мг

Мг

Мг

В главе 8 мы изучали В-корневые подгруппы на аффинных сферических С-многообразиях. Мы предложили конструкцию стандартных В-корневых подгрупп на аффинном сферическом С-многообразии X, которая обобщает известную конструкцию Т-корневых подгрупп на аффинных торических Т-многообразиях. Используя эту конструкцию, в случае орисферического X мы получили полное описание всех С-корневых подгрупп на X и доказали, что всякий С-инвариантный простой дивизор в X может быть соединён с открытой С-орбитой при помощи действия подходящей стандартной В-кориевой подгруппы. Мы получили полное описание всех В-корневых подгрупп на определённом В-инвариантном открытом подмножестве в X и нашли достаточные условия для продолжения В-корневых подгрупп с этого открытого подмножества. В качестве приложения дня произвольного аффинного сферического С-многообразия X (не обязательно орисферического) мы доказали, что всякий С-инвариантный простой дивизор в X может быть соединён с открытой С-орбитой в X при помощи действия подходящей В-корневой подгруппы.

Изучение В-корневых подгрупп на (не обязательно аффинных) сферических С-многообразиях представляется очень интересной и перспективной областью будущих исследований, которая, как мы надеемся, поможет в понимании груш: автоморфизмов таких многообразий.

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