Bruhat Tits buildings and representations of reductive p-adic groups

Bruhat Tits buildings and representations of reductive p-adic groups Maarten Solleveld Radboud Universiteit Nijmegen joint work with Ralf Meyer 26 November 2013

Starting point Let G be a reductive p-adic group and G rss be the set of regular semisimple elements of G. Let V be a complex vector space and π : G Aut(V ) an admissible representation. Theorem (Harish-Chandra) For g G rss the operator π(g) has a well-defined trace tr π (g). The function tr π : G rss C is locally constant. Goal of talk Sketch a proof with the Bruhat Tits building which: provides explicit neighborhoods of constancy; also applies to representations over other coefficient fields.

Reductive p-adic groups Let F be a local non-archimedean field of residual characteristic p. Let G be a reductive algebraic group defined over F. Then G = G(F) is called a reductive p-adic group. Examples GL n (Q p ), SL n (Q p ), O n ( Fq ((t)) ), Sp n ( Fq ((t)) ), U n ( F/F) where [ F : F] = 2 Such groups are totally disconnected and admit many compact open subgroups, for example ( 1 + p 2 Z p p 5 ) Z p Z p 1 + p 2 GL Z n (Q p ). p

Examples of affine buildings

The Bruhat Tits building of G The Bruhat Tits building B(G) = B(G, F ) is a metric G-space, on which the center Z(G) acts trivially. The isotropy group G x of any x B(G) is compact modulo center.

The Bruhat Tits building of G The Bruhat Tits building B(G) = B(G, F ) is a metric G-space, on which the center Z(G) acts trivially. The isotropy group G x of any x B(G) is compact modulo center. An apartment of an affine building is a Euclidean space A, which is usually associated to some torus T G. A building can be constructed by glueing copies of an apartment in an intricate way.

The Bruhat Tits building of G Let T be a maximal F -split torus of G, with apartment A T. B(G, F ) = G A T / where encodes precisely the isotropy groups G x for x A T.

The Bruhat Tits building of G Let T be a maximal F -split torus of G, with apartment A T. B(G, F ) = G A T / where encodes precisely the isotropy groups G x for x A T. P - minimal parabolic subgroup of G containing T P = Z G (T )U with U the unipotent radical of P K - good maximal compact subgroup of G, which fixes a special vertex x A T Iwasawa decomposition G = PK = UZ G (T )K B(G, F ) = UZ G (T )K A T = U A T (group version) (building version)

action of semisimple elements Possibilities for t T : t does not lie in a compact mod Z(G) subgroup. Then t acts as a translation on A T and it fixes no point of B(G, F); t lies in a compact mod Z(G) subgroup. Then t fixes A T pointwise.

action of semisimple elements Let T be a maximal split torus in G Every such T stabilizes an apartment A T of B(G, F) example of t T not compact mod center in GL 2 (Q 3 ): A T

action of semisimple elements Let T be a maximal split torus in G Every such T stabilizes an apartment A T of B(G, F) example of t T compact mod center in GL 2 (Q 3 ): A T fixed by t

action of semisimple elements Let T be a nonsplit maximal torus, e.g. T = { ( a b 3b a ) (a, b) Q 2 3 \ {(0, 0)} } GL 2 (Q 3 ) Every element of T is compact,

action of semisimple elements Let T be a nonsplit maximal torus, e.g. T = { ( ) a b 3b a (a, b) Q 2 3 \ {(0, 0)} } GL 2 (Q 3 ) Every element of T is compact, but g = ( 0 3 1 0 ) does not fix any apartment of B(GL 2, Q 3 ), it acts as a reflection:

action of semisimple elements Let T be a nonsplit maximal torus, e.g. T = { ( a b 3b a ) (a, b) Q 2 3 \ {(0, 0)} } GL 2 (Q 3 ) Extend the field to Q 3 [ 3], then T splits and g = ( 0 3 1 0 ) fixes an apartment of B(GL 2, Q 3 [ 3]) A T

action of unipotent elements a unipotent element u 1 does not fix any apartment of the Bruhat Tits building example: u = ( 1 x 0 1 ) GL 2(Q 3 ): fixed by u

action of unipotent elements a unipotent element u 1 does not fix any apartment of the Bruhat Tits building example: ū = ( 1 0 x 1 ) GL 2(Q 3 ): fixed by u

action of unipotent elements a unipotent element u 1 does not fix any apartment of the Bruhat Tits building example: u GL 2 (Q 2 ):

Regular semisimple elements Let T be a maximal split torus of G = G(F). Theorem For compact t T the following are equivalent: t is regular, that is, dim(z G (t)) is minimal; the distance from B(G, F) t to A T is finite; B(G, F) t /T is compact; there exists a neighborhood H of t in T, such that B(G, F) h = B(G, F) t for all h H. Also true when T is not split, but then one has to consider A T via a field extension of F

Singular depth Φ(G, T ): root system of (G, T ) over an algebraic closure F for α Φ(G, T ) : sd α (t) := val F (α(t) 1) t commutes with the root subgroup U α (F) sd α (t) =

Singular depth Φ(G, T ): root system of (G, T ) over an algebraic closure F for α Φ(G, T ) : sd α (t) := val F (α(t) 1) t commutes with the root subgroup U α (F) sd α (t) = Definition The singular depth of t T is sd(t) := max sd α(t) α Φ(G,T ) Q { } This measures how close to singular t is t is regular sd(t) <

Fixed points in the building The singular depth of t T provides an upper bound for B(G) t example: t = ( 4 0 0 1 ) GL 2(Q 3 ) sd(t) = val 3 (α(t) 1) = val 3 (3) = 1

Fixed points in the building The singular depth of t T provides an upper bound for B(G) t example: t = ( 4 0 0 1 ) GL 2(Q 3 ) sd(t) = val 3 (α(t) 1) = val 3 (3) = 1 A T fixed by t sd (t) = 1

Fixed points in the building The singular depth of t T provides an upper bound for B(G) t example: t = ( 10 0 0 1 ) GL 2(Q 3 ) sd(t) = val 3 (α(t) 1) = val 3 (9) = 2 A T fixed by t sd (t) = 2

Comparison of fixed points ht(φ): height of the longest root of Φ(G, T ) Theorem Let t T, a maximal torus of G. Suppose that g T and val F (α(g) 1) > ht(φ)sd(t) for all α Φ(G, T ) Then B(G) tg = B(G) t.

Representations of p-adic groups Recall: we wanted to study traces of admissible G-representations, and use the action of G on B(G) to show that such a traces is a locally constant function on regular semisimple elements.

Representations of p-adic groups Let V be a vector space over an arbitrary field L. Definition A representation G Aut L (V ) is smooth if for all v V there exists a compact open subgroup K G which fixes v. Example G = GL n (Q p ), V = C, π(g) = ( det(g) Qp ) i 3 Warning: the action of GL n (Q p ) on Q n p is not smooth! Rep L (G): category of smooth G-reps over L Definition (π, V ) Rep L (G) is called admissible if dim L (V K ) < for all compact open subgroups K G. All irreducible smooth representations are admissible.

Traces of admissible representations Hecke algebra of G From now on we assume char(l) p and we fix a L-valued Haar measure µ on G H(G, L): the vector space Cc (G, L) endowed with the convolution product (with respect to µ) Traces Every admissible G-rep (π, V ) is also a H(G)-module For f C c (K\G/K) H(G) dim L (im π(f )) dim L V K < so π(f ) has a well-defined trace tr π (f ) L

Traces of admissible representations We would like to regard the distribution tr π : H(G, L) L as a function on G Definition tr π (g) = c if there exists a neighborhood H of g in G s.t. tr π (f ) = c f dµ for all f Cc (H) This does not always work: if u G is unipotent, conjugacy-invariance of traces forces which is (usually) infinite H tr π (u) = tr π (1) = dim V

B(G) is the product of simplicial complexes B(G i ) with G i G almost simple A polysimplex in B(G) is the product of simplices in these factors B(G) (n) := {n-dimensional polysimplices of B(G)} This is a discrete G-space. We fix an orientation for each polysimplex. C n (B(G); Z) := free Z-module with basis B(G) (n)

A simple resolution Let V Rep L (G). C n (B(G); Z) Z V is G-representation with g(σ v) := sgn σ (g)gσ gv sgn σ detects whether g matches the orientations of σ and gσ

A simple resolution Let V Rep L (G). C n (B(G); Z) Z V is G-representation with g(σ v) := sgn σ (g)gσ gv sgn σ detects whether g matches the orientations of σ and gσ C (B(G); Z) Z V is a differential complex with G-equivariant differential n (σ v) = ( n σ) v and augmentation 0 : C 0 (B(G)) Z V V, 0 (x v) = v Because B(G) is contractible, so is V C (B(G); Z) V

A simple resolution Lemma (Bernstein) Suppose that char(l) = 0 and that Z(G) is compact. Then V C 0 (B(G)) Z V C dim B(G) (B(G)) Z V is a projective resolution in Rep L (G).

Lemma (Bernstein) A simple resolution Suppose that char(l) = 0 and that Z(G) is compact. Then V C 0 (B(G)) Z V C dim B(G) (B(G)) Z V is a projective resolution in Rep L (G). Proof of projectivity Let {σ i } be representatives for the G-orbits in B(G) (n). As G-representations C n (B(G)) Z V = i indg G σi (sgn σi V ) Rep L (G σi ) is semisimple because char(f) = 0 and G σi is compact. By Frobenius reciprocity ind G G σi (sgn σi V ) is projective in Rep L (G).

Some groups from Bruhat Tits theory O F - ring of integers of F π F O F - uniformizer Example x B(GL n (F )) fixed by GL n (O F )), e N U (e) x = ker ( GL n (O F )) GL n (O F /π e+1 F O F ) ) = 1+π e+1 F (M n (O F )) Schneider and Stuhler generalized this to pro-p-groups U σ (e) G and all polysimplices σ of B(G) Properties U gσ (e) = gu σ (e) g 1 U τ (e) U σ (e) if τ σ for all

An acyclic differential complex Let Σ B(G) be a polysimplicial subcomplex C n (e) (Σ; V ) := σ Σ (n) Zσ Z V U(e) σ n maps C n (e) (Σ; V ) to C (e) n 1 (Σ; V ) because U(e) τ U σ (e) if τ σ

An acyclic differential complex Let Σ B(G) be a polysimplicial subcomplex C n (e) (Σ; V ) := σ Σ (n) Zσ Z V U(e) σ n maps C n (e) (Σ; V ) to C (e) n 1 (Σ; V ) because U(e) τ U σ (e) if τ σ Theorem Let Σ B(G) be convex and recall that char(l) p. H n ( C (e) (Σ; V ), ) = { 0 n > 0 U(e) x Σ (0) V x n = 0 The proof depends on geometric properties of affine buildings, and on the idempotents µ(u σ (e) ) 1 1 (e) U H(G, L) σ

The Schneider Stuhler resolution C n (e) (B(G); V ) is a subrepresentation of C n (B(G)) Z V because U gσ (e) = gu σ (e) g 1 We say that V has level e if V = V U (e) x x B(G) (0) Theorem (due to Schneider Stuhler for L = C) Suppose that V has level e. V C (e) 0 (B(G); V ) C (e) dim B(G) (B(G); V ) is a resolution in Rep L (G). It is projective if char(l) = 0 and of finite type if V is admissible.

Traces Let (π, V ) be an admissible G-rep of level e. We want to make the distribution tr π into a locally constant function on G rss. With Jacquet restriction one can reduce this task to elements of a compact (modulo center) subgroup K G. The previous acyclic complexes lead to an expression of tr π on K: tr π (g) = lim ( 1) dim σ sgn σ (g)tr (π(g) ) V U(e) σ, Σ σ Σ,gσ=σ where the limit runs over all K-stable, convex finite subcomplexes Σ B(G)

Traces Let (π, V ) be an admissible G-rep of level e. We want to make the distribution tr π into a locally constant function on G rss. With Jacquet restriction one can reduce this task to elements of a compact (modulo center) subgroup K G. When g K is regular semisimple, one can find a K-stable, convex finite subcomplex Σ B(G) such that tr π (g) = σ Σ,gσ=σ ( 1) dim σ sgn σ (g)tr (π(g) ) V U(e) σ This implies that tr π is locally constant near g Crucial step: nearby elements of a torus have the same fixed points in B(G)

Conclusion Let V be a vector space over a field L whose characteristic is not p. Let π : G Aut L (V ) be an admissible representation of level e, so V = x B(G) V U(e) x Theorem Let g T G be regular semisimple. The operator π(g) has a well-defined trace tr π (g). The function tr π : G rss L is locally constant. If T contains a maximal split torus S and x A S, then tr π is constant on gu max(sd(g),e) x

Outlook Theorem Let π be an admissible G-representation. For g G rss the operator π(g) has a well-defined trace tr π (g). The function tr π : G rss L is locally constant. Harish-Chandra: with harmonic analysis (L = C) Meyer Solleveld: via the Bruhat Tits building of G. When char(f) = 0, Harish-Chandra also showed that tr π extends to a locally integrable function G C and that tr π (f ) = tr π (g)f (g)dµ(g) f H(G; C) G With our methods one can try to generalize this to char(f) > 0