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# Geometry of B×B-orbit closures in equivariant embeddings

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*in*Advances in Mathematics 216(2):626-646 · December 2007

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DOI: 10.1016/j.aim.2007.06.001 · Source: arXiv

Cite this publicationAbstract

Let X denote an equivariant embedding of a connected reductive group G over an algebraically closed field k. Let B denote a Borel subgroup of G and let Z denote a B×B-orbit closure in X. When the characteristic of k is positive and X is projective we prove that Z is globally F-regular. As a consequence, Z is normal and Cohen–Macaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that Z has rational singularities. This extends earlier results by the second author and M. Brion.

- ... We call these cells and their closures generalised (and opposite generalised) Schubert cells and varieties. They have many properties of the classical Schubert cells and varieties: normality, Cohen-Macaulay property (see for example [4, 5, 7] for more details). We study these varieties and their intersections, that we call generalised Richardson varieties, as well as the images of these varieties under morphisms of G-embeddings. ...... Using this technique, Rittatore [19] obtained regularity results for all G-embeddings, in particular the Cohen-Macaulay property. Brion and Polo [4], Brion and Thomsen [5] and He and Thomsen [7] also obtained regularity results for B × B-orbit closures in group embeddings. For rational projective homogeneous spaces, Knutson, Lam and Speyer [9] proved that in G/P (with P a parabolic subgroup containing B) the projections of Richardson varieties are all the compatibly split subvarieties for the unique B-canonical splitting. ...... We introduce projected generalised Richardson varieties (see Definition 4.2) and prove the following result. The fact that projected generalised Richardson varieties are compatibly split follows from results of He and Thomsen [7]. We use techniques of Knutson, Lam and Speyer [9] to prove that these varieties are the only compatibly split subvarieties. ...Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.
- ... The Frobenius splitting statement for matrices was proved by X. He and J.F. Thomsen in [8]. The last section of this article is devoted to more explicit computations. ...ArticleFull-text available
- Dec 2014

We prove that closures of Borel conjugacy classes of $2$-nilpotent matrices have a rational resolution of singularities. - ... 4], see the end of Section 3. We also allow simple factors of the Levi of P − to be in the kernel of π. For results on B × B-orbit closures in G × G-equivariant embeddings of G we refer to [18] which uses the notions of strong and global F -regularity. The main results of this paper are contained in Section 4. In Proposition 4.1 we give the divisor of a B-semi-invariant function on G/H. ...Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay. Examples are the $G\times G$-orbits in normal reductive monoids with unit group $G$. Our class of homogeneous spaces also includes the open orbits of the well-known determinantal varieties and the varieties of (circular) complexes. We also show that all $G$-orbit closures in a spherical variety which is canonically Frobenius split are normal. Finally we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.
- ... The closures of such intersections also appear in the study of algebrogeometric properties of G. In the joint work [16] of He and Thomsen, it was proved that in positive characteristics, there exists a Frobenius splitting on G which compatibly splits all the nonempty intersections of the closures of B × B-orbits and B − × B − -orbits in G. In particular, all such closures are weakly normal and reduced. ...Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let be the wonderful compactification of G. For a fixed pair (B,B−) of opposite Borel subgroups of G, we look at intersections of Lusztig’s G-stable pieces and the B−×B-orbits in , as well as intersections of B×B-orbits and B−×B−-orbits in . We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of Ḡ.
- Article
- Oct 2018
- TRANSFORM GROUPS

We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height 2 in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type A or the element has rank 2. - Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups.
- Article
- Apr 2007
- J ALGEBRA

Let X be the wonderful compactification of a semisimple adjoint algebraic group. Extending the standard monomials on the flag variety, Chirivi and Maffei constructed a basis of the space of global sections on X that is compatible with all closures of GxG-orbits in X. We show that this basis is also compatible with all BxB-orbit closures by defining subsets using only combinatorics of the underlying paths. Furthermore, we construct standard monomials on X that have properties similar to classical standard monomials. - Article
- Dec 2010
- ADV MATH

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification. - Article
- Jun 2010
- ADV MATH

We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q-divisor Δ such that −KX−Δ is ample and (X,Δ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-Q-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi–Yau” condition. We also prove a Kawamata–Viehweg vanishing theorem for globally F-regular pairs. - We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen Macaulay.
- Article
- Dec 2011
- Represent Theor

We present a geometric proof of Bernstein's second adjointness for a reductive $p$-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a "specialization" map between spaces of functions on various varieties with $G\times G$ action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of functors lead to the second adjointness. We also get a formula for the "specialization" map expressing it as a composition of the orishperic transform and inverse intertwining operator; a parallel result for $D$-modules was obtained in arXiv:0902.1493. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra, generalizing a result by Opdam. - We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph \Gamma of G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (\Gamma, \Gamma, \id).

- Article
- Apr 2007
- J ALGEBRA

Let X be the wonderful compactification of a semisimple adjoint algebraic group. Extending the standard monomials on the flag variety, Chirivi and Maffei constructed a basis of the space of global sections on X that is compatible with all closures of GxG-orbits in X. We show that this basis is also compatible with all BxB-orbit closures by defining subsets using only combinatorics of the underlying paths. Furthermore, we construct standard monomials on X that have properties similar to classical standard monomials. - Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst F -rational rings have rational singularities
- Jan 1987
- 61-90

- A Ramanathans
- K E Smith

A, Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. Haute Etudes Sci. Publ. Math. 65 (1987), 61–90. [S] K. E. Smith, F -rational rings have rational singularities, Amer. J. Math. 119 (1997), 159–180. - Frobenius Splittings Methods in Geometry and Representation Theory The behaviour at infinity of the Bruhat decomposition
- Jan 1998
- 73-137

- M Brion
- S Kumar

M. Brion and S. Kumar, Frobenius Splittings Methods in Geometry and Representation Theory, Progress in Mathematics (2004), Birkhäuser, Boston. [B] M. Brion, The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), 137–174. - Intersection cohomology of B × B-orbits closures in group compactifications Frobenius splitting of equivariant closures of regular conjugacy classes, math
- Jan 2002
- 71-111

[Sp] T. A. Springer, Intersection cohomology of B × B-orbits closures in group compactifications, J. Alg. 258 (2002), 71–111. [T] J.F. Thomsen, Frobenius splitting of equivariant closures of regular conjugacy classes, math.AG/0502114. - Institute for Advanced Study USA E-mail address: hugo@math.ias.edu Institut for matematiske fag
- Mathematics School

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA E-mail address: hugo@math.ias.edu Institut for matematiske fag, Aarhus Universitet, 8000Århus8000Århus C, Denmark E-mail address: funch@imf.au.dk - Global F -regularity of Schubert varieties with applications to D-modules, math.AG/0402052. [R] A, Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals
- Jan 1987
- 65-61

- N Lauritzen
- U R Pedersen
- J F Thomsen

N. Lauritzen, U. R. Pedersen and J. F. Thomsen, Global F -regularity of Schubert varieties with applications to D-modules, math.AG/0402052. [R] A, Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. HautesÉtudesHautes´HautesÉtudes Sci. Publ. Math. 65 (1987), 61–90. - This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of F-rationality and a treatment of F-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $\S6$ and is then applied in $\S7$ to prove that both tight closure and F-regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in $\S6$ that for a reduced ring R essentially of finite type over an excellent local ring of characteristic p, if c is not in any minimal prime of R and Rc is regular, then c has a power that is a test element. It is shown in $\S7$ that if S is a flat R-algebra with regular fibers and R is F-regular then S is F-regular. The general problem of showing that tight closure commutes with smooth base change remains open, but is reduced here to showing that tight closure commutes with localization.
- Article
- Jan 1998
- COMMENT MATH HELV

For a connected reductive group G and a Borel subgroup B, we study the closures of double classes BgB in a (G G) (G \times G) -equivariant "regular" compactification of G. We show that these closures [`(BgB)] \overline {BgB} intersect properly all (G G) (G \times G) -orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all [`(BgB)] \overline {BgB} are singular in codimension two exactly. We deduce this from more general results on B-orbits in a spherical homogeneous space G/H; they lead to formulas for homology classes of H-orbit closures in G/B, in terms of Schubert cycles. - Jan 2000
- 97-126

- M Brion
- P Polo

M. Brion and P. Polo, Large Schubert Varieties, Represent. Theory 4 (2000), 97–126.- Article
- Sep 2004
- Am J Math

Let G denote a connected reductive algebraic group over an algebraically closed field k and let X denote a projective G x G-equivariant embedding of G. The large Schubert varieties in X are the closures of the double cosets BgB, where B denotes a Borel subgroup of G, and g is in G. We prove that these varieties are globally F-regular in positive characteristic, resp. of globally F-regular type in characteristic 0. As a consequence, the large Schubert varieties are normal and - Article
- Dec 2002

Consider a flag variety Fl over an algebraically closed field, and a subvariety V whose cycle class is a multiplicity--free sum of Schubert cycles. We show that V is arithmetically normal and Cohen--Macaulay, in the projective embedding given by any ample invertible sheaf on Fl. - Article
- Jun 2003
- J ALGEBRAIC GEOM

Introduction Let G be a connected semisimple, simply connected linear algebraic group over an algebraically closed field k and B be a Borel subgroup in G. If w = (P 1 , . . . , P n ) is a sequence of minimal parabolic subgroups containing B, we may form the quotient Zw = Pw/B , where Pw = P n and B acts on Pw from the right via (p 1 , . . . , p n )(b 1 , . . . , b n ) = (p 1 b 1 , b -1 1 p 2 b 2 , b -1 2 p 3 b 3 , . . . , b -1 n-1 p n b n ). The quotient Zw is inductively a sequence of P -bundles with natural sections starting with the P -bundle P 1 /B (over a point). The product map Pw G induces a proper morphism #w : Zw G/B whose image is a Schubert variety in G/B. For "reduced" sequences w the morphism #w is birational and equal to the celebrated Demazure desingularization of the Schubert variety #w (Zw ). In general we call Zw the Bott-Samelson variety associated with w .The construction of Zw originates in the papers [1][2][3] of Bott-Samelson, Demazure and - Article
- Jan 2003
- J ALGEBRA

this paper concern the intersection cohomology of the closures of the B B-orbits. Examples of such closures are the "large Schubert varieties," the closures in X of the double cosets BwB in G - It is shown that the tight closure of a submodule in a Artinian module is the same as its finitistic tight closure, when the modules are graded over a finitely generated N-graded ring over a perfect field. As a corollary, it is deduced that for such a graded ring, strong and weak F-regularity are equivalent. As another application, the following conjecture of Hochster and Huneke is proved: Let (R, m) be a finitely generated N-graded ring over a held with unique homogeneous maximal ideal m, then R is (weakly) F-regular if and only if R-m is (weakly) F-regular.
- . It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational. A key point in the proof is a characterization of F-rational local rings as those Cohen-Macaulay local rings (R; m) in which the local cohomology module H d m (R) (where d is the dimension of R) have no submodules stable under the natural action of the Frobenius map. An analog for finitely generated algebras over a field of characteristic zero is developed, which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over C , without reference to a desingularization. With the development of the theory of tight closure by M. Hochster and C. Huneke [HH1], a natural question arose. What information does this powerful new tool provide about the structure of the singularities of an algebraic variety? The main theorem of this paper is the following: Theorem 3.1. If a...
- Article
- Dec 2000

this paper concern the intersection cohomology of the closures of the B Theta B-orbits. Examples of such closures are the "large Schubert varieties", the closures in X of the double cosets BwB in G - Article
- May 1999
- Represent Theor

For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically van der Kallen's filtration of the algebra of regular functions on $B$. We also construct a degeneration of the flag variety $G/B$ embedded diagonally in $G/B\times G/B$, into a union of Schubert varieties. This leads to formulae for the class of the diagonal in $T$-equivariant $K$-theory of $G/B\times G/B$, where $T$ is a maximal torus of $B$. - Article
- Mar 2005
- P LOND MATH SOC

Let G denote a connected semisimple and simply connected algebraic group over an algebraically closed field k of positive characteristic and let g denote a regular element of G. Let X denote any equivariant embedding of G. We prove that the closure of the conjugacy class of g within X is normal and Cohen–Macaulay. Moreover, when X is smooth we prove that this closure is a local complete intersection. As a consequence, the closure of the unipotent variety within X shares the same geometric properties. 2000 Mathematics Subject Classification 14M17 (primary), 13A35 (secondary). - Let $G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $\bar{G}$ of $G$ into finite many $G$-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We will show that the closure is a disjoint union of some $G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $G$-orbits.
- Article
- Mar 2004
- J AM MATH SOC

We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.