Flag varieties as equivariant compactifications of G_a^n
aa r X i v : . [ m a t h . AG ] M a r FLAG VARIETIES AS EQUIVARIANT COMPACTIFICATIONS OF G na IVAN V. ARZHANTSEV
Abstract.
Let G be a semisimple affine algebraic group and P a parabolic subgroup of G .We classify all flag varieties G/P which admit an action of the commutative unipotent group G na with an open orbit. Introduction
Let G be a connected semisimple affine algebraic group of adjoint type over an algebraicallyclosed field of characteristic zero, and P be a parabolic subgroup of G . The homogeneousspace G/P is called a (generalized) flag variety. Recall that
G/P is complete and the action ofthe unipotent radical P − u of the opposite parabolic subgroup P − on G/P by left multiplicationis generically transitive. The open orbit O of this action is called the big Schubert cell on G/P . Since O is isomorphic to the affine space A n , where n = dim G/P , every flag varietymay be regarded as a compactification of an affine space.Notice that the affine space A n has a structure of the vector group, or, equivalently, ofthe commutative unipotent affine algebraic group G na . We say that a complete variety X ofdimension n is an equivariant compactification of the group G na , if there exists a regular action G na × X → X with a dense open orbit. A systematic study of equivariant compactificationsof the group G na was initiated by B. Hassett and Yu. Tschinkel in [4], see also [10] and [1].In this note we address the question whether a flag variety G/P may be realized as anequivariant compactification of G na . Clearly, this is the case when the group P − u , or, equiv-alently, the group P u is commutative. It is a classical result that the connected component e G of the automorphism group of the variety G/P is a semisimple group of adjoint type, and
G/P = e G/Q for some parabolic subgroup Q ⊂ e G . In most cases the group e G coincides with G , and all exceptions are well known, see [6], [7, Theorem 7.1], [12, page 118], [3, Section 2].If e G = G , we say that ( e G, Q ) is the covering pair of the exceptional pair (
G, P ). For asimple group G , the exceptional pairs are (PSp(2 r ) , P ), (SO(2 r + 1) , P r ) and ( G , P ) withthe covering pairs (PSL(2 r ) , P ), (PSO(2 r + 2) , P r +1 ) and (SO(7) , P ) respectively, where P H denotes the quotient of the group H by its center, and P i is the maximal parabolic subgroupassociated with the i th simple root. It turns out that for a simple group G the condition e G = G implies that the unipotent radical Q u is commutative and P u is not. In particular,in this case G/P is an equivariant compactification of G na . Our main result states that theseare the only possible cases. Date : November 6, 2018.2010
Mathematics Subject Classification.
Primary 14M15; Secondary 14L30.
Key words and phrases.
Semisimple group, parabolic subgroup, flag variety, automorphism.Supported by RFBR grants 09-01-00648-a, 09-01-90416-Ukr-f-a, and the Deligne 2004 Balzan prize inmathematics.
Theorem 1.
Let G be a connected semisimple group of adjoint type and P a parabolic sub-group of G . Then the flag variety G/P is an equivariant compactification of G na if and onlyif for every pair ( G ( i ) , P ( i ) ) , where G ( i ) is a simple component of G and P ( i ) = G ( i ) ∩ P , oneof the following conditions holds: the unipotent radical P ( i ) u is commutative; the pair ( G ( i ) , P ( i ) ) is exceptional. For convenience of the reader, we list all pairs (
G, P ), where G is a simple group (up tolocal isomorphism) and P is a parabolic subgroup with a commutative unipotent radical:(SL( r + 1) , P i ) , i = 1 , . . . , r ; (SO(2 r + 1) , P ); (Sp(2 r ) , P r );(SO(2 r ) , P i ) , i = 1 , r − , r ; ( E , P i ) , i = 1 ,
6; ( E , P ) , see [9, Section 2]. The simple roots { α , . . . , α r } are indexed as in [2, Planches I-IX]. Notethat the unipotent radical of P i is commutative if and only if the simple root α i occurs in thehighest root ρ with coefficient 1, see [9, Lemma 2.2]. Another equivalent condition is thatthe fundamental weight ω i of the dual group G ∨ is minuscule, i.e., the weight system of thesimple G ∨ -module V ( ω i ) with the highest weight ω i coincides with the orbit W ω i of the Weylgroup W . 1. Proof of Theorem 1
If the unipotent radical P − u is commutative, then the action of P − u on G/P by left mul-tiplication is the desired generically transitive G na -action, see, for example, [5, pp. 22-24].The same arguments work when for the connected component e G of the automorphism groupAut( G/P ) one has
G/P = e G/Q and the unipotent radical Q − u is commutative. Since G/P ∼ = G (1) /P (1) × . . . × G ( k ) /P ( k ) , where G (1) , . . . , G ( k ) are the simple components of the group G , the group e G is isomorphic tothe direct product g G (1) × . . . × g G ( k ) , cf. [8, Chapter 4]. Moreover, Q u ∼ = Q (1) u × . . . × Q ( k ) u with Q ( i ) = g G ( i ) ∩ Q , Thus the group Q − u is commutative if and only if for every pair ( G ( i ) , P ( i ) )either P ( i ) u is commutative or the pair ( G ( i ) , P ( i ) ) is exceptional.Conversely, assume that G/P admits a generically transitive G na -action. One may identify G na with a commutative unipotent subgroup H of e G , and the flag variety G/P with e G/Q ,where Q is a parabolic subgroup of e G .Let T ⊂ B be a maximal torus and a Borel subgroup of the group e G such that B ⊆ Q .Consider the root system Φ of the tangent algebra g = Lie( e G ) defined by the torus T , itsdecomposition Φ = Φ + ∪ Φ − into positive and negative roots associated with B , the set ofsimple roots ∆ ⊆ Φ + , ∆ = { α , . . . , α r } , and the root decomposition g = M β ∈ Φ − g β ⊕ t ⊕ M β ∈ Φ + g β , where t = Lie( T ) is a Cartan subalgebra in g and g β = { x ∈ g : [ y, x ] = β ( y ) x for all y ∈ t } is the root subspace. Set q = Lie( Q ) and ∆ Q = { α ∈ ∆ : g − α * q } . For every root LAG VARIETIES AS COMPACTIFICATIONS OF G na β = a α + . . . + a r α r define deg( β ) = P α i ∈ ∆ P a i . This gives a Z -grading on the Lie algebra g : g = M k ∈ Z g k , where t ⊆ g and g β ⊆ g k with k = deg( β ) . In particular, q = M k ≥ g k and q − u = M k< g k . Assume that the unipotent radical Q − u is not commutative, and consider g β ⊆ [ q − u , q − u ]. Forevery x ∈ g β \ { } there exist z ′ ∈ g β ′ ⊆ q − u and z ′′ ∈ g β ′′ ⊆ q − u such that x = [ z ′ , z ′′ ]. In thiscase deg( z ′ ) > deg( x ) and deg( z ′′ ) > deg( x ).Since the subgroup H acts on e G/Q with an open orbit, one may conjugate H and assumethat the H -orbit of the point eQ is open in e G/Q . This implies g = q ⊕ h , where h = Lie( H ).On the other hand, g = q ⊕ q − u . So every element y ∈ h may be (uniquely) written as y = y + y , where y ∈ q , y ∈ q − u , and the linear map h → q − u , y y , is bijective. Take theelements y, y ′ , y ′′ ∈ h with y = x, y ′ = z ′ , y ′′ = z ′′ . Since the subgroup H is commutative,one has [ y ′ , y ′′ ] = 0. Thus[ y ′ + y ′ , y ′′ + y ′′ ] = [ y ′ , y ′′ ] + [ y ′ , y ′′ ] + [ y ′ , y ′′ ] + [ y ′ , y ′′ ] = 0 . But [ y ′ , y ′′ ] = x and [ y ′ , y ′′ ] + [ y ′ , y ′′ ] + [ y ′ , y ′′ ] ∈ M k> deg( x ) g k . This contradiction shows that the group Q − u is commutative. As we have seen, the lattercondition means that for every pair ( G ( i ) , P ( i ) ) either the unipotent radical P ( i ) u is commutativeor the pair ( G ( i ) , P ( i ) ) is exceptional. The proof of Theorem 1 is completed.2. Concluding remarks
If a flag variety
G/P is an equivariant compactification of G na , then it is natural to askfor a classification of all generically transitive G na -actions on G/P up to equivariant isomor-phism. Consider the projective space P n ∼ = SL( n + 1) /P . In [4], a correspondence betweenequivalence classes of generically transitive G na -actions on P n and isomorphism classes oflocal (associative, commutative) algebras of dimension n + 1 was established. This corre-spondence together with classification results from [11] yields that for n ≥ G na -actions on P n is infinite, see [4, Section 3].On the contrary, a generically transitive G na -action on the non-degenerate projective quadric Q n ∼ = SO( n + 2) /P is unique [10, Theorem 4]. It would be interesting to study the sameproblem for the Grassmannians Gr( k, r + 1) ∼ = SL( r + 1) /P k , where 2 ≤ k ≤ r − Acknowledgement
The author is indebted to N.A. Vavilov for a discussion which results in this note. Thanksare also due to D.A. Timashev and M. Zaidenberg for their interest and valuable comments.
I.V. ARZHANTSEV
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