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Geometric Structure for the Principal Series of a Split Reductive P-Adic Group with Connected Centre

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Geometric Structure for the Principal Series of a Split Reductive P-Adic Group with Connected Centre Geometric structure for the principal series of a split reductive p-adic group with connected centre Aubert, Anne-Marie and Baum, Paul and Plymen, Roger and Solleveld, Maarten 2015 MIMS EPrint: 2015.77 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 GEOMETRIC STRUCTURE FOR THE PRINCIPAL SERIES OF A SPLIT REDUCTIVE p-ADIC GROUP WITH CONNECTED CENTRE ANNE-MARIE AUBERT, PAUL BAUM, ROGER PLYMEN, AND MAARTEN SOLLEVELD Abstract. Let G be a split reductive p-adic group with connected cen- tre. We show that each Bernstein block in the principal series of G admits a definite geometric structure, namely that of an extended quo- tient. For the Iwahori-spherical block, this extended quotient has the form T ==W where T is a maximal torus in the Langlands dual group of G and W is the Weyl group of G. Contents 1. Introduction 1 2. Extended quotient 3 3. The group W s as a Weyl group 4 4. Comparison of different parameters 5 4.1. Varieties of Borel subgroups 5 4.2. Enhanced Langlands parameters 7 4.3. Affine Springer parameters 8 5. Structure theorem 10 6. Correcting cocharacters and L-packets 13 References 15 1. Introduction Let G be a split reductive p-adic group with connected centre, and let G = G_ denote the Langlands dual group. Then G is a complex reductive group. Let T be a maximal torus in G and let W be the common Weyl group of G and G. We can form the quotient variety T=W and, familiar from noncommutative geometry [Kha, p.77], the noncommu- tative quotient algebra O(T ) o W: Date: August 1, 2015. 2010 Mathematics Subject Classification. 20G05, 22E50. Key words and phrases. reductive p-adic group, representation theory, geometric struc- ture, local Langlands conjecture. The second author was partially supported by NSF grant DMS-0701184. 1 2 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD Within periodic cyclic homology (a noncommutative version of de Rham theory) there is a canonical isomorphism ∗ HP∗(O(T ) o W ) ' H (T ==W ; C) where T ==W denotes the extended quotient of T by W , see x2. In this sense, the extended quotient T ==W , a complex algebraic variety, is a more concrete version of the noncommutative quotient algebra O(T ) o W . Returning to the p-adic group G, let Irr(G)i denote the subset of the smooth dual Irr(G) comprising all the irreducible smooth Iwahori-spherical representations of G. We prove in this article that there is a continuous bijective map, satisfying several constraints, as follows: T ==W ' Irr(G)i: We note that there is nothing in the classical representation theory of G to indicate that Irr(G)i admits such a geometric structure. Nevertheless, such a structure was conjectured by the present authors in [ABPS], and so this article is a confirmation of that conjecture, for the single point i in the Bernstein spectrum of G. We prove, more generally, that, subject to constraints itemized in [ABPS], and subject to the Condition 3.1 on the residual characteristic, there is a continuous bijective map T s==W s ' Irr(G)s for each point s in the Bernstein spectrum for the principal series of G, see Theorem 5.2. Here, T s and W s are the complex torus and the finite group attached to s. This, too, is a confirmation of the geometric conjecture in [ABPS]. Let WF denote the Weil group of F . A Langlands parameter Φ for the principal series of G should have Φ(WF ) contained in a maximal torus of G. In particular, it should suffice to consider parameters Φ such that Φ WF ab ∼ × factors through WF = F , that is, such that Φ factors as follows: × Φ: WF × SL2(C) ! F × SL2(C) ! G: Such a parameter is enhanced in the following way. Let ρ be an irreducible representation of the component group of the centralizer of the image of Φ: ρ 2 Irr π0(ZG(im Φ)): The pair (Φ; ρ) will be called an enhanced Langlands parameter. We rely on Reeder's classification of the constituents of a given principal series representation of G, see [Ree, Theorem 1, p.101-102]. Reeder's the- orem amounts to a local Langlands correspondence for the principal series of G. Reeder uses only enhanced Langlands parameters with a particular geometric origin, namely those which occur in the homology of a certain variety of Borel subgroups of G. This condition is essential, see, for exam- ple, the discussion, in [ABP2], of the Iwahori-spherical representations of the exceptional group G2. In Theorem 4.7 we show how to replace the enhanced Langlands param- eters of this kind, namely those of geometric origin, by the affine Springer GEOMETRIC STRUCTURE 3 parameters defined in x4.3. These affine Springer parameters are defined in terms of data attached to the complex reductive group G { in this sense, the affine Springer parameters are independent of the cardinality q of the residue field of F . The scene is now set for us to prove the first theorem of geometric structure, namely Theorem 5.1, from which our main structure theorem, Theorem 5.2 follows. We also relate our basic structure theorem with L-packets in the principal series of G, see Theorem 6.2. An earlier, less precise version of our conjecture was formulated in [ABP1]. That version was proven in [Sol1] for Bernstein components which are de- scribed nicely by affine Hecke algebras. These include the principal series of split groups (with possibly disconnected centre), symplectic and orthogonal groups and also inner forms of GLn. Acknowledgements. Thanks to Mark Reeder for drawing our attention to the article of Kato [Kat]. We thank Joseph Bernstein, David Kazhdan, George Lusztig, and David Vogan for enlightening comments and discus- sions. 2. Extended quotient Let Γ be a finite group acting on a complex affine variety X by automor- phisms, Γ × X ! X: The quotient variety X=Γ is obtained by collapsing each orbit to a point. For x 2 X; Γx denotes the stabilizer group of x: Γx = fγ 2 Γ: γx = xg: Let c(Γx) denote the set of conjugacy classes of Γx. The extended quotient is obtained from X=Γ by replacing the orbit of x by c(Γx). This is done as follows: Set Xe = f(γ; x) 2 Γ × X : γx = xg. It is an affine variety and a subvariety of Γ × X. The group Γ acts on Xe: Γ × Xe ! Xe α(γ; x) = (αγα−1; αx); α 2 Γ; (γ; x) 2 X:e The extended quotient, denoted X==Γ, is X=e Γ. Thus the extended quotient X==Γ is the usual quotient for the action of Γ on Xe. The projection Xe ! X; (γ; x) 7! x is Γ-equivariant and so passes to quotient spaces to give a morphism of affine varieties ρ: X==Γ ! X=Γ: This map will be referred to as the projection of the extended quotient onto the ordinary quotient. The inclusion X,! Xe x 7! (e; x) e = identity element of Γ 4 A.-M. AUBERT, P. BAUM, R. PLYMEN, AND M. SOLLEVELD is Γ-equivariant and so passes to quotient spaces to give an inclusion of affine varieties X=Γ ,! X==Γ. This article will be dominated by extended quotients of the form T ==W or, more generally, extended quotients of the form T s==W s. 3. The group W s as a Weyl group Let G be a connected reductive p-adic group over F , which is F -split and has connected centre. Let T be a F -split maximal torus in G. Let G, T denote the Langlands dual groups of G, T . The principal series consists of all G-representations that are obtained with parabolic induction from characters of T . We will suppose that the residual characteristic p of F satisfies the hypothesis in [Roc, p. 379], for all reductive subgroups H ⊂ G containing T : Condition 3.1. If the root system R(H; T ) is irreducible, then the restric- tion on the residual characteristic p of F is as follows: • for type An p > n + 1 • for types Bn;Cn;Dn p 6= 2 • for type F4 p 6= 2; 3 • for types G2;E6 p 6= 2; 3; 5 • for types E7;E8 p 6= 2; 3; 5; 7: If R(H; T ) is reducible, one excludes primes attached to each of its irreducible factors. Since R(H; T ) is a subset of R(G; T ) =∼ R(G; T )_, these conditions are fulfilled when they hold for R(G; T ). We denote the collection of all Bernstein components of G of the form s = [T ; χ]G by B(G; T ) and call these the Bernstein components in the principal series. The union [ Irr(G; T ) := Irr(G)s s2B(G;T ) is by definition the set of all irreducible subquotients of principal series representations of G. Choose a uniformizer $F 2 F . There is a bijection t 7! ν between points in T and unramified characters of T , determined by the relation ν(λ($F )) = λ(t) ∗ [T ,χ] where λ 2 X∗(T ) = X (T ). The space Irr(T ) T is in bijection with T via t 7! ν 7! χ⊗ν. Hence Bernstein's torus T s is isomorphic to T .
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