A GENERALIZATION OF FULTON S CONJECTURE FOR AN EMBEDDED SUBGROUP

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1 A GENERALIZATION OF FULTON S CONJECTURE FOR AN EMBEDDED SUBGROUP JOSHUA KIERS Abstract. In this paper we extend a result on the relationship between intersection theory of general flag varieties and invariant theory of certain representations. This already generalized a conjecture of Fulton on the nature of the Littlewood-Richardson coefficients. The main result of this paper is that the mantra multiplicity one intersection theory gives rise to rigidity in representation theory continues to hold when the intersections in question are of Schubert classes in two different flag varieties, in the situation where one reductive algebraic group is embedded in another in a particular fashion. 1. Introduction In this paper we extend the main result of [BKR12] on the relationship between intersection theory of general flag varieties and invariant theory of certain representations. This already generalized a conjecture of Fulton (first proved by Knutson-Tao-Woodward in [KTW04]) on the nature of the Littlewood-Richardson coefficients. The main result of this paper is that the mantra multiplicity one intersection theory gives rise to rigidity in representation theory continues to hold when the intersections in question are of Schubert classes in two different flag varieties, in the situation where one reductive algebraic group is embedded in another in a particular fashion. The layout of this paper is very similar to that of [BKR12], and no essentially new proofs appear here. Let G Ă p G be connected reductive algebraic groups over C. Fix a maximal torus H and Borel subgroup B Ą H of G, and let p H Ă p B be the same for p G, also satisfying H Ă p H, B Ă p B. Let λ be a one parameter subgroup λ : C Ñ H satisfying 9λ : d dt λptqˇˇˇt 1 P h`; i.e., λ is G-dominant. Define the Kempf s parabolic P pλq associated to λ to be P pλq : tg P G : lim tñ0 λptqgλptq1 exists in Gu, and define p P pλq Ă p G similarly. One easily verifies that B Ă P pλq, p B Ă p P pλq, and P pλq p P pλq X G. Thus Lemma 1.1. The induced map is an inclusion. φ λ : G{P pλq ãñ p G{ p P pλq We will often write P, p P for P pλq, p P pλq, respectively. Define associated Levi subgroups Lpλq : tg P G : lim tñ0 λptqgλptq1 gu and, analogously, p Lpλq. These we will also refer to by L and p L. The Lie algebras of H, B, L, P, G will be denoted by h, b, l, p, g, and those of p H, p B,... by p h, p b,... Definition 1.2. Say a pair w P W P, pw P x W pp is Levi-movable if the map T 9e pg{p q Ñ T 9epG{P q T 9e plλ w q T 9ep p G{ p P q T 9e p p l p Λ pw q induced by id ˆ dφ λ is an isomorphism for generic pl, p lq P L ˆ pl. Let symbols d w pw denote the structure coefficients in φ λ pr p X pw sq ÿ wpw P d w pw rx ws. An equivalent definition for L-movability can be given (see [RR11, Proposition 2.3]): 1

2 2 JOSHUA KIERS Proposition 1.3. Suppose w P W P, pw P x W pp. Set w _ w 0 ww P 0, where w 0, w P 0 are the longest elements of W, W P, respectively. Then the pair w _, pw is Levi-movable if and only if d w pw 0 and pi pχ pw χ w q p 9 λq 0, where χ w P h, pχ pw P p h are linear functionals given by ÿ χ w : β and pχ pw : and i : h ãñ p h is the inclusion. Let φ λ βppr`zr` P qxw1 R` denote the usual pull-back in singular cohomology: φ λ : H p p G{ p P ; Zq Ñ H pg{p ; Zq ÿ βpp p R`z p R` xp qx pw1 p R` As introduced in [RR11], there is a deformed pull-back ring homomorphism φ d λ : H p p G{ p P ; Z, d 0 q Ñ H pg{p ; Z, d 0 q defined by φ d λ pr X p pw sq ÿ c w pw rx ws, wpw P where c w pw dw pw if pw_, pwq is Levi-movable and c w pw 0 otherwise. The following is our main theorem: Theorem 1.4. Let w, pw P W P ˆ xw pp be such that pw s d 0 rx w s rx e s P H pg{p q. φ d λ β, Then for any n ě 1, VL dim pnχ w q b V pl pnpχ pwq L The geometry For w P W P, pw P W x pp, let Q w (resp. Q p pw ) be the stabilizer of X w ( X p pw ) under the left action of G ( G) p on G{P ( G{ p P p ). Define Y w : Q w C w Q w 9w, and define Z w to be the smooth locus of X w. It is easy to see that X w Ą Z w Ą Y w Ą C w, and the analogous statement in G{ p P p holds as well. Set X : pg ˆB X w q ˆ pg p ˆ p pb X pw q, and define analogous spaces Z, Y, C by replacing X w, X p pw with the relevant varieties. Define X to be the schematic fibre-product of δ and m, X X ˆm m δ G{P G{P ˆ pg{ p P, where δ pid, φ λ q and m is the multiplication map rg, xs, rpg, pxs ÞÑ gx, pgpx. Set theoretically, X!ḡ 1, ḡ 2, z P G{B ˆ pg{ B p ˆ pg{ P p ) z P φ λ pg 1 X w q X g 2X p pw. One similarly defines schemes Z, Y, C using Z, Y, C. If (1) pw s d 0 rx w s drx e s P H pg{p q, φ d λ then the generic number of points in the intersection φ λ pg 1 X w q X g 2 p X pw is d. If d 1 the natural projection map π : X Ñ G{B ˆ pg{ p B is birational. Now we introduce a lemma similar in spirit to [BKR12, Lemma 4.2]. The following setup is essentially the same. Let Y Ă X be irreducible smooth varieties, Y locally closed in X. Suppose X has a transitive action by a connected

3 A GENERALIZATION OF FULTON S CONJECTURE FOR AN EMBEDDED SUBGROUP 3 linear algebraic group G, and suppose H is an algebraic subgroup fixing Y. For any y P Y, define φ y : G Ñ X by g ÞÑ gy. Then for any g P G, there is an induced tangent space map Because Y is H-stable, there is an induced map dφ pg,yq : T g G Ñ T gy X. Φ pg,yq : TḡpG{Hq Ñ T gy X{T gy pgy q. One easily checks that Φ pg,yq Φ pgh,h1 yq if h P H, so for each equivalence class rg, ys P G ˆH Y the map Φ rg,ys is well-defined. The transitivity of the G-action implies that the maps Φ rg,ys are surjective. Suppose a rg, zs, rpg, pzs P Z. Define x gz, px pgpz. In particular, px φ λ pxq. Consider the following diagram of maps of tangent spaces T a Z dπ T g pg{bq T pg p p G{ p Bq (2) d ˆm T x pg{p q Ψ rg,zsˆψ r pg, pzs T x pg{p q T x pgz w q T pxp p G{ p P q T px ppg p Z pw q, where the bottom horizontal map is the canonical projection in the first factor and dφ λ followed by the canonical projection in the second factor. Lemma 2.1. Diagram (2) commutes. In fact, it is a fibre-product diagram. Proof. A generic curve through a in Z may be expressed as prgptq, ˆ zptqs, rpgptq, pzptqsq, where gp0q g, etc. The image under dπ of this curve s initial velocity is the initial velocity of gptq, gptq y. Its further image under Ψ rg,zs ˆ Ψ rpg,pzs is the pair of projections in the respective quotients of the initial velocities of gptqzptq and pgptqpzptq. Note that pgptqpzptq φ λ pgptqzptqq for all t. Therefore the curve s image via the down and across composition agrees and the diagram commutes. ˆ That T a Z is a subspace of (i.e., includes into) the fibre-product is clear since, for a curve gptq, gptq y through pg, pgq in T g pg{bq T pg pg{ p Bq p and corresponding xptq through x in G{P, the curve prgptq, zptqs, rpgptq, pzptqsq can be uniquely recovered via zptq : gptq1 xptq, pzptq : pgptq1 φ λ pxptqq. Counting dimensions, dim Z dim G{P ` dimpg ˆB Z w q ` dimpg p ˆ p pb Z pw q dim G{P ` dim G{ p P p dim G{P ` dim G{B ` dim Z w ` dim G{ p B p ` dim Z p pw dim G{P dim G{ p P p dim G{P ` dim G{B ` dim G{ p B p pdim G{P dim Z w q dim G{ p P p dim Z pw p, so T a Z has the correct dimension and the result follows. We will go on to use this Lemma to better understand the relationship between Y, Z, and R. First, however, let us see how Y is related to the representation theory. 3. Representation theory Similar to Y, set Y 1 : pg ˆQw Y w q ˆ pg p ˆ p pqxw Y pw q. Then define the scheme Y 1 as the analogous fibre-product: Y 1 Y 1 m δ G{P G{P ˆ pg{ p P. Consider the following commutative diagram, inducing a map p : Y Ñ Y 1 :

4 4 JOSHUA KIERS Y Y pˆˆp Y 1 Y 1 m m where p and ˆp are the natural projections. Because δ G{P G{P ˆ pg{ p P, pˆp Y Y 1 π π 1 G{B ˆ pg{ p B G{Q w ˆ pg{ p Q pw commutes (the bottom horizontal map is the natural projection in each factor), so does p Y Y 1 (3) π π 1 G{B ˆ pg{ p B G{Q w ˆ pg{ p Q pw. Both π and π 1 are dominant maps, so [BKR12, Lemma 4.1] tells us that the ramification divisor R of π is exactly p pr 1 q, where R 1 is the ramification divisor of π 1. In particular, Lemma 3.1. For each n ě 1, as G-modules. Now, define the following (diagonal) P -variety and define a map H 0 py, OpnRq Y q» H 0 py 1, OpnR 1 qq P `P {w1 Q w w X P ˆ pp { pw1 p Q pw pw X p P ψ : G ˆP P Ñ Y 1, where, g, p, p ı ÞÑ rgpw1, wp s, rgp pw1, pw P p s. One checks that ψ is independent of choices. Furthermore, it is easy to check that ψ is a bijection. Proposition 3.2. Suppose φ d λ pw s d 0 rx w s drx e s P H pg{p q for some d ą 0. Then H 0 py, OpnRq Y q G» `V L pnpχ w χ 1 qq b V pl pnpχ pw q L Proof. Let T P T 9e pg{p q, T pp T 9e p p G{ p P q, T w T 9e pλ w q, and T pw T 9e p p Λ pw q. For any pg, p, pq P G ˆP P, let a ψprg, p, psq. There is the fibre-product diagram dψ dπ T pg,p,pq pg ˆP Pq T a Y 1 T gpw1pg{q w q T gp pw1p p G{ p Q pw q d ˆm T gp pg{p q T gp pg{p q T pgpw1 gp Y w q T gp pp G{ p P p q T gp ppgp pw1 Y p pw q,

5 A GENERALIZATION OF FULTON S CONJECTURE FOR AN EMBEDDED SUBGROUP 5 in exact analogy to (2). There are P -equivariant isomorphisms and P {w1 Q w w X P ˆ T P» P ˆw1 Q wwxp T P pp { pw1 p Q pw pw X p P ˆ T pp» p P ˆ pw1 p Qxw pwx p P T p P given by p p, vq ÞÑ pp, p1 vq in both cases, cf. [BK06, Definition 5]. Therefore there exist maps and P ˆ T P Ñ P {w1 Q w w X P ˆ T P» P ˆw1 Q wwxp T P Ñ P ˆw1 Q wwxp pt P {T w q P ˆ T P Ñ p P { pw1 p Q pw pw X p P ˆ T P ãñ p P { pw1 p Q pw pw X p P ˆ T pp» p P ˆ pw1 p Qxw pwx p P T pp Ñ p P ˆ pw1 p Qxw pwx p P pt pp {T pw q. By [BKR12, Lemma 4.1], the ramification divisor ψ1 pr 1 q in G ˆP P is the same as the ramification divisor of the bundle map G ˆP pp ˆ T P q Ñ G ˆP pp ˆw1 Q wwxp pt P {T w qq G ˆP p p P ˆ pw1 p Qxw pwx p P pt pp {T pw qq over G ˆP P. Setting M L P pχ w χ 1 q b L pp pχ pw q, a line bundle over P, we conclude (cf. the discussion surrounding [BK06, Lemma 6] and [BKR12, Proposition 6.2]) that Opφ1 prqq is G-isomorphic to G ˆP M as line bundles over G ˆP P. Therefore for any n, H 0 py, OpnRqq G» H 0 py 1, OpnR 1 qq G» H 0 pg ˆP P, G ˆP M bn q G» H 0 pp, M bn q P. Now, set L L{pw1 Q w w X Lq ˆ pl{p pw1 p Q pw pw X p Lq. Then, following [BK06, Theorem 15, Remark 31(a)], it also holds that H 0 pp, M bn q P» H 0 pl, pm L q bn q L, from which the result follows. Having identified H 0 py, OpnRqq G in terms of irreducible representations of L, we now wish to show that, if d 1 in (1) for a pair w, pw, then dim H 0 py, OpnRqq G 1 for every n. However, we must first perform an analysis of the tangent spaces of subvarieties of the relevant flag varieties in order to understand the relationship between Y, Z, and R. Let w P W P. Then as H-modules, 4. Tangent spaces T 9e pw1 X w q» γpw1 R`XR Recall the following statement proved in [BKR12, Lemma 7.3]: Lemma 4.1. Suppose v ÝÑ β w P W P. As H-modules, Equivalently, as H-modules, Now, 9 λ satisfies T 9v px w q» T 9e pv1 X w q» γpr`xvr γpv1 R`XR g γ. g γ gβ. αpλq 9 α P pp q αpλq 9 ą α P z pp q. Therefore λ 9 may play the role of x P in [BKR12, Section 7]. As a direct sum of H-eigenspaces, T 9e pg{p q βpr`zr` l g γ gv1 β. T 9e pg{p qβ.

6 6 JOSHUA KIERS Define, for any j P Z, V j : β P R`zR` l βp 9 λq j T 9e pg{p qβ. Note that V j p0q if j ď 0 or j ą m 0 : max β tβpλqu. Define V j pzq : V j X T 9e pzq for any H-stable subvariety of G{P containing 9e. Then T 9e pzq V j pzq j as H-modules. If Z is only zplq-stable, the above decomposition is a zplq-module decomposition. Recall an important theorem from [BKR12, Theorem 7.4] (see also [Res, Proposition 3]): Theorem 4.2. Given that u β ÝÑ w P W P and β is not simple, there exists j such that dim V j pu1 Z w q dim V j pw1 Z w q. In exact parallel, and one may define T 9e p p G{ p P q pv j : pβp p R`z p R` p l T 9e p p G{ p P q p β, pβ P p R`z p R` p l pβp 9 λq j T 9e p p G{ p P q p β. The analogue to the above theorem holds in this setting as well. Because dφ λ : T 9e pg{p q ãñ T 9e pg{ p P p q is an H-equivariant inclusion, it follows that for any β P R, the restriction of dφ λ satisfies dφ λ : T 9e pg{p q β ãñ T 9ep G{ p P p q pβ. pβ ˇ β h In particular, then, dφ λ : V j ãñ p V j for each j P Z. 5. Main theorem Proposition 5.1. Suppose w, pw P W P ˆ xw pp are Levi-movable, and suppose pw s d 0 rx w s drx e s P H pg{p q for some d ą 0. There exists a closed subset A of Z of codimension at least 2 such that φ d λ ZzY Ă R Y A. Proof. It suffices to show that, for all irreducible codimension 1 subsets B of Z, For this it suffices to show that, if u, pu satisfy either B X pzzrq H. (4) u ÝÑ β w or pu ÝÑ β pw, with β not a simple root, then C u ˆ pu pc X pzzrq H. Assume there exists u, pu as in (4) and a prg, zs, rpg, pzsq such that (5) a P C u ˆ pu pc X pzzrq. As the pair pw, pwq is L-movable, the map ψ : T 9e pg{p q Ñ T 9epG{P q T 9e plλ w q T 9ep p G{ p P q T 9e p p l p Λ pw q induced by id ˆ dφ λ is an isomorphism for generic l P L, p l P L. p The latter decomposes (since lλ w is zplq-stable, and similarly for p lλ pw ) as m0 V j pg{p q m0 pv j pg{ p p P q V j plλ w q pv j p p lλ, pw q j 1 j 1

7 A GENERALIZATION OF FULTON S CONJECTURE FOR AN EMBEDDED SUBGROUP 7 and since ψ preserves H-weight spaces with the same 9 λ action, for each j we must have V j pg{p q» V jpg{p q V j plλ w q p V j p p G{ p P q pv j p p l p Λ pw q. Consequently, for each j, F j pg{p q» F jpg{p q F j plλ w q p F j p p G{ p P q pf j p p l p Λ pw q, and the same holds replacing l, p l with pl, p p l for any p P P, p P P p due to the P -stability of F j (resp. P p -stability of Fj p ). Now, under assumption (5), let x : gz φ1 λ ppgpzq. Since G fixes the intersection in question, we may assume x 9e. By regularity and Lemma 2.1, T 9e pg{p q» T 9epG{P q T 9e pgz w q T 9ep p G{ p P q T 9e ppg p Z pw q. Since ep gz gbup for suitable b P B, g pu1b1 for some p P P and T 9e pgz w q T ppu1 9e Z w q. T 9e ppg Z p pw q T pppu1 9e Z p pw q for some p P P p. In particular, for any j, is injective; therefore F j Ñ F j F j ppu1 Z w q p Fj pf j pppu1 p Z pw q dim F j ď dim F j dim F j ppu1 Z w q ` dim p F j dim p F j pppu1 p Z pw q. Of course dim F j ppu1 Z w q dim F j pu1 Z w q and dim p F j pppu1 p Z pw q dim p F j ppu1 p Z pw q. Now, one can easily verify that for each j the inequalities dim F j pw1 Z w q ď dim F j pu1 Z w q and hold in general. Furthermore, by Theorem 4.2, there exists a j such that dim p F j p pw1 p Z pw q ď dim p F j ppu1 p Z pw q dim F j pw1 Z w q dim F j pu1 Z w q or dim p F j p pw1 p Z pw q dim p F j ppu1 p Z pw q, depending on which assertion in (4) is satisfied. In either case, it follows that (for the above j) Similarly, dim F pw1 j Z w q ` dim F p j p pw1 Z p pw q ă dim F pu1 j Z w q ` dim F p ppu1 j Z p pw q. The contradiction arises from dim F j ď dim F j dim F ppu1 j Z w q ` dim F p j dim F p pppu1 j Z p pw q dim F j dim F pu1 j Z w q ` dim F p j dim F p ppu1 j Z p pw q ă dim F j dim F pw1 j Z w q ` dim F p j dim F p j p pw1 Z p pw q dim F j dim F j plλ w q ` dim F p j dim F p j p p lλ pw q dim F j, which concludes the proof. Finally we are in a position to prove the main theorem, whose proof is identical to [BKR12, Theorem 8.2]: Theorem 5.2. Suppose w, pw P W P ˆ xw pp are Levi-movable, and suppose pw s d 0 rx w s rx e s P H pg{p q. φ d λ Then for any n ě 1, dimpv L pnχ w q b V pl pnpχ pw qq L 1. By Proposition 3.2 it suffices to show that H 0 py, OpnRq Y q G is one-dimensional for all n ě 0. By Proposition 5.1, H 0 py, OpnRq Y q ãñ H 0 pz, OpmpnqRqq, where mpnq is some positive integer depending on n. By [BKR12, Proposition 3.1], H 0 pz, OpmRqq» C for each m ě 1. As the constants belong to H 0 py, OpnRq Y q G, the chain of inclusions gives the desired result. C ãñ H 0 py, OpnRq Y q G ãñ H 0 py, OpnRq Y q ãñ H 0 pz, OpmpnqRqq» C

8 8 JOSHUA KIERS References [BK06] P. Belkale and S. Kumar, Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math. 166 (2006), no. 1, Ò5 [BKR12] P. Belkale, S. Kumar, and N. Ressayre, A generalization of Fulton s conjecture for arbitrary groups, Math. Ann. 354 (2012), no. 2, Ò1, 2, 4, 5, 6, 7 [KTW04] A. Knutson, T. Tao, and C. Woodward, The honeycomb model of GL npcq tensor products. II. Puzzles determine the facets of the Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004), no. 1, Ò1 [Res] N. Ressayre, Distribution on homogeneous spaces and Belkale-Kumar s product, Transform. Groups. To appear. Ò6 [RR11] N. Ressayre and E. Richmond, Branching Schubert calculus and the Belkale-Kumar product on cohomology, Proc. Amer. Math. Soc. 139 (2011), no. 3, Ò1, 2 Department of Mathematics, University of North Carolina, Chapel Hill, NC jokiers@live.unc.edu