Dual Blobs and Plancherel Formulas

132(1), 2004, p. 55–80

DUAL BLOBS AND PLANCHEREL FORMULAS by Ju-Lee Kim

Abstract. — Letkbe ap-adic field. Let Gbe the group ofk-rational points of a connected reductive groupGdefined overk, and letgbe its Lie algebra. Under certain hypotheses on G and k, wequantify the tempered dualGb of Gvia the Plancherel formula ong, using some character expansions. This involves matching spectral de- composition factors of the Plancherel formulas on gand G. As a consequence, we prove that any tempered representation contains a good minimalK-type; we extend this result to irreducible admissible representations.

R´esum´e(Blobs duaux et formule de Plancherel). — Soientkun corpsp-adique,Gun groupe r´eductif connexe d´efini surk,Gson groupe de pointsk-rationnels etgl’alg`ebre de Lie deG. Sous certaines hypoth`eses, nousquantifionsle dual temp´er´eGbdeGpar la formule de Plancherel surg, en utilisant des d´eveloppements en caract`eres. Pour cela, il faut en particulier mettre en correspondance les facteurs de la d´ecomposition spectrale de la formule de Plancherel surget surG. Comme cons´equence, nous d´emontrons que toute repr´esentation temp´er´ee contient un bonK-type minimal ; nous ´etendons aussi ce r´esultat aux repr´esentations admissibles irr´eductibles.

Texte re¸cu le 15 mars 2002, r´evis´e les 22 novembre 2002 et 7 f´evrier 2003, accept´e le 28 f´evrier 2003.

Ju-Lee Kim, School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Department of Mathematics Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 • E-mail :julee@math.uic.edu

2000 Mathematics Subject Classification. — 22E50, 22E35, 20G25.

Key words and phrases. — Representations, p-adic groups, Plancherel formula, character expansions.

Research partially supported by NSF grants DMS 9970454 and 9729992.

BULLETIN DE LA SOCI ´ET ´E MATH ´EMATIQUE DE FRANCE 0037-9484/2004/55/$ 5.00 c

Soci´et´e Math´ematique de France

Introduction

Letkbe ap-adic field,Ga connected reductive group defined overk, andG the group ofk-rational points ofG. LetGebe the set of the equivalence classes of irreducible admissible representations of G. To study G, representationse of compact open subgroups of G have been useful. In particular, Moy and Prasad studied open compact subgroups coming from the theory of the Bruhat- Tits buildingB(G) ofG. They introduced (unrefined) minimalK-types of the form s:= (Gx,%, χ), whereGx,% is an open compact subgroup ofGassociated to (x, %) ∈ B(G)×R_≥0, and χ is an irreducible representation of Gx,% of a certain type. If we introduce weak associativity, an equivalence relation of minimalK-types (see Definition 2.2.1), we can partitionGeas follows

Ge= [◦ s∈SK

Ges,

where Ges is the set of (π, Vπ)∈Gecontaining a minimalK-type weakly associ- ated tos, andSKis the set of equivalence classes of weakly associated minimal K-types.

Let Gb be the tempered dual of G. Recall that Gb is the support of the Plancherel measures in the unitary dual ofG. Let bgbe the unitary dual of g.

Thenbgis also a tempered dual ofg. In this paper, we “quantify” thePlancherel integral over each Gbs := Gb∩Ges in terms of the Plancherel integral over an appropriateG-domain in the Lie algebragofG, when the residue characteristic ofk is large. Based on the Kirillov theory of compactp-adic groups [10], such a quantification first appeared in the work of Howe [11]. To start with, we observe that from Plancherel formulas ongandG, we have

(Pl)

Z

b g'g

fb(X)dX =f(0) = Z

Gb

Θπ(f◦log)dπ

for anyf ∈C_c^∞(bg)'C_c^∞(g) supported in a small neighborhood of 0 (here, we identify gandbgvia Pontrjagin duality). Now, we can refine this equality: we find an equality between spectral decomposition factors of each side (see 1.4.5 and 3.2.6) of (Pl), where spectral components of the right hand side are param- eterized by certain minimal K-types, and those of the left hand side by some relatedG-domains inbg.

To be more precise, from now on, we fix agoodminimalK-types= (Gx,%, χ).

Note that the depth ofsis%. A minimalK-type isgood when itsdual blob(the dual coset realizingχ) is good (see [3] or 1.2.2, 3.2.1). LetS be the dual blob ofs, and letGeS ⊂Gesbe the set of (π, Vπ)∈Gescontaining goodK-types weakly associated toS(in this paper, we introduce three weak associativities: between minimalK-types (2.2.1), between good cosets (1.4.1), between goodK-types and good cosets (3.2.3)). Then we match the Plancherel integral overGbS :=Gb∩GeS

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with the Fourier transform of some distribution supported on a G-domaing_S in gcoming fromS. Roughly speaking, g_S is the G-orbit of good dual blobs weakly associated tos(see 1.4.3). In particular, for two good minimalK-types s⁰ and s⁰⁰ with dual blobsS⁰ and S⁰⁰, ifGes⁰∩Ges⁰⁰ =∅, theng_S0 ∩g_S00 =∅. Moreover, g= ∪g^◦ _S0 (see 1.4.5) where S⁰ runs over weakly associated classes of good cosets. We remark that the depth of any (π, Vπ) ∈ GeS is the same as the depth % of s, and the depth of g_S (see Definition 1.2.3) is −%. Let g%:=S

x∈B(G)gx,%. When% >0, we prove that for anyf ∈C_c^∞(g%), (1)

Z

g_S

fb(X)dX = Z

GbS

Θπ(f◦log)dπ.

Summing overS in weakly associated classes of good cosets, we see Z

g

f(Xb )dX = X

S

Z

g_S

fb(X)dX (2)

= X

S

Z

GbS

Θπ(f◦log)dπ = Z

Gb

Θπ(f◦log)dπ,

which leads to the proof that every tempered representation contains a good minimal K-type (see Theorem 4.5.1) when the residue characteristic is large.

Under the same hypothesis, we also show that any irreducible admissible rep- resentation contains a good minimalK-type. This fact has been already proved (see [13, 2.4.10]) using more tools from the theory of buildings. Although we still use such tools quite a bit, this work is just an initial step to approach the problem on the exhaustion of the types constructed in [12]. In this sense, the purpose of this paper is rather a piecewise quantification, i.e., a matching of spectral decomposition factors as in (1) (see Theorem 3.3.1). However, one can hope that this analytic approach might be more fruitful for generalization.

To prove the equality in (1), we regard both sides of (1) as distributions on C_c^∞(g%). Here, the domain where (1) holds is restricted to g%, however, this is large enough to single out Ges from Ge in the following sense: there are f ∈ C_c^∞(g%) such that Θπ(f ◦log) 6= 0 implies π ∈ Ges. On the other hand, C_c^∞(g%⁺) (here, g_%⁺ = S

s>%gs) can not isolate better than the set of depth %representations which strictly contains Ges. In the case of depth zero representations, it is not possible to distinguish a single class ofK-types by dual cosets. Hence replacingg_S and GeS by a G-domain g0 and the set of all depth zero representationsGe0, we prove an analogous equality onC_c^∞(g0⁺). However, this is good enough to prove that any tempered representation contains a good minimalK-type.

We approach this problem via various character expansions and homogeneity results. When%= 0, by the work of Waldspurger and DeBacker (see [18], [5]), we know that the Harish-Chandra-Howe local character expansion is valid on

the setg0⁺of topologically nilpotent elements ing. We show that the distribu- tions in (1) are in the span of Fourier transforms of nilpotent orbital integrals when restricted tog0⁺. Then we match two distributions using Gelfand-Graev functions as test functions as in [4].

If % > 0, the Harish-Chandra-Howe expansions are not enough, because the set g%⁺ where they hold is not big enough. Hence we use Γ-asymptotic expansions of representations in GeS. In [13], F. Murnaghan and the au- thor proved that such expansions are valid on the G-domain g%. More pre- cisely, assume that π contains a good type coming from a good element Γ of depth −% (see [3] or 1.2.1–1.2.2 for definition). Denote the set of G-orbits whose closure contains Γ by O(Γ). Then we can express the character distri- bution Θπas a linear combination of Fourier transforms of orbital integralsµO

with O ∈ O(Γ). That is, if π is an irreducible admissible representation of G containing (Gx,%, χ), and ifχ is realized by a good element Γ, we prove that there arecO(π)∈Cindexed byO(Γ) such that

Θπ(expX) = X

O∈O(Γ)

cO(π)µcO(X),

and this expansion is valid on g%∩greg. Then we show that the distributions in (1) are in the span of Fourier transforms ofµOwithO ∈ O(Γ) when restricted to g%, and we match two distributions using some test functions found in [13].

In the first section, we recall some basic definitions related to this work, and state the hypotheses that we use at various places of this paper. We also define the weak associativity of good cosets, which induces a partition of g accordingly (§1.4). Then we show that this partition induces a spectral decomposition as in (2) (see Lemma 1.4.5). In Section 2, we define the weak associativity of minimalK-types, which induces a partition of G. In Section 3,e we relate the partitions on gand Ge found in the first two sections. Section 4 is basically devoted to proving the equality in (1) (see Theorem 3.3.1). As an application, in the end of Section 4, we prove that any tempered representation contains a good minimalK-type; we also discuss the extension of this result to irreducible admissible representations. As a corollary, we also get a new spectral decomposition of the delta distribution on Gwhere each decomposition factor is parameterized by the G-orbit of a good dual blob.

Acknowledgments. — This work was motivated from conversations with R. Howe. I would like to thank him for helpful discussions. I would like to thank A.-M. Aubert, S. DeBacker and F. Murnaghan for helpful comments.

I thank the Institute for Advanced Study for its friendly and stimulating atmosphere.

Notation and Conventions. — Letk be a p-adic field (a finite extension of Q_p) with residue field F_pn. Let ν = νk be the valuation on k such that

o

ν(k^×) =Z. Letk be an algebraic closure ofk. For an extension fieldE of k, letνE be the valuation onE extendingν. We will just writeν forνE. LetOE

be the ring of integers of E with prime ideal pE. Let Λ be a fixed additive character of ksuch that ΛO_k6= 1 and Λp_k= 1.

Let Gbe a connected reductive group defined overk, and G(E) the group ofE-rational points ofG. We denote byGthe group ofk-rational points ofG. Denote the Lie algebras ofGandG(E) bygandg(E), respectively. Denote by g^∗ and g^∗(E) the linear duals ofg andg(E) respectively. We writegand g^∗ for the vector space of k-rational points of gand g^∗ respectively. In general, we use bold charactersH,M,N,etc. to denote algebraic groups defined overk, corresponding Roman charactersH, MandN to denote the groups ofk-points, andh,mandnto denote the Lie algebras ofH, M andN.

Denote the set of regular elements ingbygreg. LetN be the set of nilpotent elements in g. There are different notions of nilpotency. However, since we assume that char(k) = 0, those notions are all the same. We refer to [14], [6]

for more discussion of this.

If X is a topological space with a Borel measure dx and if Y is a subset ofX, volX(Y) denotes the volume ofY with respect to dx.

For any subsetS ing, we denote by [S] the characteristic function ofS, and by−S the set{−s|s∈S}. Forg∈G, ^gZ denotesgZg⁻¹and forH ⊂G, ^HS denotes{^gZ|Z ∈S, g∈H}.

LetGe be the set of equivalence classes of irreducible admissible representa- tions of G. LetGb be the subset of Ge which consists of equivalence classes of tempered representations of G.

1. Good cosets and g

1.1. Moy-Prasad filtrations. — For a finite extensionE ofk, letB(G, E) denote the extended Bruhat-Tits building ofGoverE. For a maximal torusT in G, if it splits overE, let A(T, E) be the corresponding apartment over E.

It is known that if E⁰ is a tamely ramified Galois extension ofE,B(G, E) can be embedded into B(G, E⁰) and its image is equal to the set of the Galois fixed points in B(G, E⁰) (see [17, 5.11] or [16]). Moreover, we have

A(T, E) =A(T, E⁰)∩ B(G, E).

We let A(T, k) :=A(T, E)∩ B(G, k).

RegardingGas a group defined overE, Moy and Prasad associateg(E)x,r

andG(E)x,|r|to (x, r)∈ B(G, E)×Rwith respect to the valuation normalized as follows [14]: let E^u be the maximal unramified extension ofE, and L the minimal extension ofE^u over whichGsplits. Then the valuation used by Moy and Prasad maps L^× ontoZ.

In a similar way, with respect to our normalized valuation ν, we can define filtrations in g(E) and G(E). Then our g(E)x,r and G(E)x,r correspond to g(E)x,e`r and G(E)x,e`r of Moy and Prasad, wheree = e(E/k) is the rami- fication index of E over k and ` = [L : E<sup>u</sup>]. Hence, if $E is a uniformizing element of E, our filtrations satisfy $Eg(E)x,r = g(E)x,r+1/e while theirs satisfy$Eg(E)x,r=g(E)x,r+`.

This normalization is chosen to have the following property [1, 1.4.1]: for a tamely ramified Galois extensionE⁰ ofE andx∈ B(G, E)⊂ B(G, E⁰), we have

g(E)x,r=g(E⁰)x,r∩g(E).

Ifr >0, we also have

G(E)x,r=G(E⁰)x,r∩G(E).

Remark 1.1.1. — Letr∈ ¹_eZ. Two pointsxandy inB(G, E) lie in the same facet if and only if

g(E)x,r=g(E)y,r and g(E)x,r⁺=g(E)y,r⁺.

1.1.2. — For simplicity, we put B(G) :=B(G, k) and gx,r :=g(k)x,r, etc. We will also use the following notation. Letr∈R. Then,

1) gx,r⁺=S

s>rgx,s and Gx,|r|⁺=S

s>|r|Gx,s, x∈ B(G);

2) g^∗_x,r=

χ∈g^∗|χ(g_x,(−r)⁺)⊂pk , x∈ B(G);

3) gr=S

x∈B(G)gx,r and gr⁺ =S

s>rgs; 4) Gr=S

x∈B(G)Gx,r and Gr⁺=S

s>rGs for r≥0.

For (x, r)∈ B(G, E)×R, we can define corresponding objects ing(E),g^∗(E) andG(E). We will denote them using (E).

1.2. Depth functions and good elements. — Recall that the depth func- tion on B(G, k)×g is a function d : B(G, k)×g→ R defined as follows: for X ∈gandx ∈ B(G, k), let d(x, X) =rbe the depth ofX in thex-filtration, that is,ris the unique real number such thatX ∈gx,r\g_x,r⁺. We also define

d(X) = sup_x∈B(G,k)d(x, X).

Note that the depth d(X) ofXis the uniquerinR∪{∞}such thatX ∈gr\gr⁺. Moreover, d is well defined and locally constant on g\ N, and it is ∞ on N (see [2, 3.3.4]). We can also define a depth function d^E over a finite exten- sionE ofk. IfEis tamely ramified, thanks to our normalization of valuation, we observe that for any x ∈ B(G, k) and X ∈ g, d(x, X) = d^E(x, X) and d(X) = d^E(X) (see [2]). Hence we may omit the superscript ‘E’ in that case.

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LetT be a maximalk-torus which splits over a tamely ramified Galois ex- tensionE ofk, and lettbe its Lie algebra. ThenT :=T(k) andt:=t(k) have the following filtrations:

• forr∈R, tr:=

Γ∈t|ν(dχ(Γ))≥rfor allχ∈X^∗(T) and

• forr >0, Tr:=

t∈T |ν(χ(t)−1)≥rfor allχ∈X^∗(T) . Note that Tr=T∩T(E)r andtr=t∩t(E)r.

Definition 1.2.1 (see [3]). — Let Tbe a maximal k-torus which splits over a tamely ramified Galois extension of k, and lettbe its Lie algebra.

1) If Γ∈tr\tr⁺, we say that Γ isof depthrwith respect toT, and we write dT(Γ) =r.

2) Let Γ∈gbe a semisimple element of depthr with respect tot. Then Γ is calledgood with respect to Tif for every rootαofGwith respect toT,dα(Γ) is either zero or has valuationr.

The depth and goodness of semisimple elements do not depend on the choice of T, and in fact dT(Γ) = d(Γ) (see [13, 2.1.2], [1]). Note that 0∈gis a good element of depth ∞.

Definition 1.2.2 (see [3], [13]). — 1) Let r <0. A coset S =X +gx,r⁺

with X ∈gx,r\g_x,r⁺ isgoodif there is a good element Γ of depth rsuch that Γ +g_x,r⁺=X+g_x,r⁺andx∈ B(C_G(Γ), k).

2) Forx∈ B(G, k),S :=gx,0is called a 0-good coset.

Definition and Remark 1.2.3. — Let S be a good coset. Define the depth d(S) ofS as

d(S) := min

X∈Sd(X).

Then if S is a 0-good coset, d(S) = 0. If S = Γ +gx,r⁺ with r < 0, for any X ∈ S, then d(S) = d(X) =rby [2, 3.3.7].

1.3. B(G⁰, k). — LetTbe a maximalk-torus inGwhich splits over a tamely ramified Galois extension E of k, and let t be its Lie algebra. Let Γ be a semisimple element in t. LetG⁰ denote the centralizerC_G(Γ) of Γ inG, andg⁰ its Lie algebra. Then (G⁰,G) forms a twisted Levi sequence, that is, G⁰(E) is an E-Levi subgroup of G(E) (see [20]). In particular, if Γ splits over k, G⁰ is a Levi subgroup of G. In general, G⁰(E) is a Levi subgroup ofG(E), and hence there is a Galois equivariant embedding ofB(G⁰, E) intoB(G, E), which in turn induces an embedding ofB(G⁰, k) intoB(G, k) (see [1,§1.9] or [20, 2.11]).

Such embeddings are unique modulo translation by elements ofX_∗(Z_G0, k)⊗R. However, the images remain the same.

Fix such an embedding i:B(G⁰, k)→ B(G, k). Then we will regardB(G⁰, k) as a subset of B(G, k) and write just x for i(x). For any x ∈ B(G⁰, k), the

associated filtrations on G⁰ := G⁰(k) and g⁰ := g⁰(k) are given as follows (see [1, 1.9.1]):

G⁰_x,r=G(E)x,r∩G⁰=Gx,r∩G⁰ forr >0, g⁰_x,r=g(E)x,r∩g⁰=gx,r∩g⁰ for anyr∈R.

For later use, we need some lemmas on good cosets. The implication of Lemma 1.3.2 on good K-types can be found in Corollary 3.2.2. We first need to recall the following. Let r ∈ R. In [6], DeBacker defined the notion of generalizedr-facetsin the affine building of a reductive group. For the purposes of this paper, it suffices to consider the set of generalizedr-facets inB(G⁰, k).

Definition 1.3.1 (see [6]). — Forx∈ B(G⁰, k), define F^∗(x) : =

y∈ B(G⁰, k)|g⁰_x,r=g⁰_y,r andg⁰_x,r+ =g⁰_y,r+

=

y∈ B(G⁰, k)|G⁰_x,|r|=G⁰_y,|r|andG⁰_x,|r|+ =G⁰_y,|r|+ , F(r) : =

F^∗(x)|x∈ B(G⁰, k) .

An element in F(r) is called ageneralizedr-facet inB(G⁰, k). ForF^∗:=F^∗(x) in F(r), define

g⁰_F∗:=g⁰_x,r, g⁰⁺_F∗:=g⁰_x,r+, G⁰_F∗:=G⁰_x,r, G⁰⁺_F∗:=G⁰_x,r+.

Lemma 1.3.2. — Letx ∈ B(G⁰, k), and letX⁰ ∈gx,r∩g⁰_r+. Then there exist g∈G⁰_x,0 andy∈ B(G⁰, k)such that

(i) Γ + ^gX⁰+gx,r⁺⊂Γ +gy,r⁺, and (ii) gx,r⁺⊂gy,r⁺ andgx,r⊃gy,r.

Proof. — SinceX⁰∈g⁰_r+, there is a nilpotent elementn⁰ ∈g⁰ such that Γ +X⁰+g⁰_x,r+= Γ +n⁰+g⁰_x,r+.

Let F^∗ ⊂ B(G⁰, k) be a maximal generalizedr-facet which contains x in the closure. From [15, 6.3], we can deduce that there is a g ∈ G⁰_x,0 such that n⁰⁰= ^gn⁰∈g⁰⁺_F∗. Then

Γ + ^gX⁰+g_x,r⁺= Γ + ^gn⁰+g_x,r⁺ = Γ +n⁰⁰+g_x,r⁺.

Recall that Γ splits over a tamely ramified Galois extensionE of k. Hence G⁰(E) is anE-Levi subgroup of G(E). Let Tbe a maximal E-split torus inG such that Γ ∈ t(E), F^∗∩ A(T, E) 6= ∅, and x ∈ A(T, E). Note that since T⊂G⁰, A(T, E)⊂ B(G⁰, E). LetC ⊂ A(T, E) be a chamber inB(G, E) such that x∈C andF^∗∩C6=∅. Choose ˜y∈C. Then

g(E)y,r˜ ⊂g(E)x,r and g(E)x,r⁺⊂g(E)y,r˜ ⁺.

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For anyz ∈F^∗∩C, g⁰_z,r+ =g⁰_F⁺∗ and g⁰(E)z,r⁺ ⊂g⁰(E)y,r˜ ⁺. Hencen⁰⁰ is an element of g⁰_F⁺∗ ⊂g⁰(E)y,r˜ ⁺. Let y be the center of mass of the Galois orbit of ˜y. Theny∈ A(T, k)⊂ B(G⁰, k). Note that

n⁰⁰∈ \

σ∈Gal(E/k)

g(E)y˜^σ,r⁺⊂g(E)y,r⁺,

g(E)_x,r⁺⊂ \

σ∈Gal(E/k)

g(E)_y˜^σ_,r⁺⊂g(E)_y,r⁺,

g(E)y,r⊂ X

σ∈Gal(E/k)

g(E)y˜^σ,r⊂g(E)x,r.

Hence y satisfies the required properties.

Remarks 1.3.3. — Let Γ be a good element and let G⁰ := C_G(Γ). Let x in B(G⁰, k). We recall some results from [13,§2].

1) If X⁰∈gx,r∩g⁰_r+, we have^Gx,0+(Γ +X⁰+g⁰_x,r+) = Γ +X⁰+gx,r⁺. 2) B(G⁰, k) ={x∈ B(G, k)|d(x,Γ) = d(Γ)}.

3) For any y∈ B(G, k)\ B(G⁰, k) and X⁰∈g⁰_r+, d(y,Γ +X⁰) is strictly less thanr.

4) Let X1,X2∈Γ +g⁰_x,r+. If ^gX1=X2 for someg∈G, theng∈CG(Γ).

Lemma 1.3.4. — Let Γ be a good element and let G⁰ := C_G(Γ). Let x in B(G⁰, k), and let X⁰ ∈ gx,r∩g⁰_r+. Let Z+gz,r⁺ be a coset in gz,r where z∈ B(G, k). If (Γ +X⁰+gx,r⁺)∩(Z+gz,r⁺)6=∅, then there existh∈Gx,0⁺

andn⁰∈g⁰_hz,r∩ N such thathz∈ B(G⁰, k)andZ+gz,r⁺=^h−1(Γ +n⁰) +gz,r⁺. Proof. — Since (Γ +X⁰+gx,r⁺)∩(Z+gz,r⁺)6=∅, by Remark 1.3.3, 1), there existh∈Gx,0⁺andY⁰ ∈X⁰+g⁰_x,r+ such that^h−1(Γ +Y⁰)∈Z+gz,r⁺. Hence Γ +Y⁰+ghz,r⁺ =^hZ+ghz,r⁺. From Remark 1.3.3, 3), hz ∈ B(G⁰, k). Since Y⁰ ∈g⁰_r+by [2, 3.2.2, 3.2.6], there is ann⁰∈g⁰_hz,r∩N such that Γ+n⁰+ghz,r⁺= Γ +Y⁰+ghz,r⁺. HenceZ+gz,r⁺=^h−1(Γ +n⁰) +gz,r⁺.

Lemma 1.3.5. — LetT be a maximalk-torus in Gwhich splits over a tamely ramified extensionE. LetΓ1,Γ2∈tbe two good semisimple elements of depthr such thatΓ1≡Γ2 (mod tr⁺). ThenCG(Γ1) =CG(Γ2). Moreover,Γ1−Γ2is in the center of the Lie algebra of CG(Γ1).

Proof. — Let x ∈ A(T, k). Then Γ1+gx,r⁺ = Γ2+gx,r⁺ is a good coset.

Applying Remark 1.3.3, 4) forX1=X2= Γ1 and Γ = Γ2, we haveCG(Γ1)⊂ CG(Γ2). Similarly,CG(Γ2)⊂CG(Γ1). Hence CG(Γ1) =CG(Γ2). The second statement follows from the first one.

Hypotheses 1.3.6. — Here we list the hypotheses used in various places of this paper. These are already used in [4], [5], [13]. We will state explicitly whenever these hypotheses are necessary.

(HB) There is a nondegenerate G-invariant symmetric bilinear form B on g such that g^∗_x,r is identified with gx,r via the map Ω : g → g^∗ defined by Ω(X)(Y) = B(X, Y).

(HGT) Every maximal k-torus Tin G splits over a tamely ramified Galois extension, and for any r∈R, any coset in tr/t_r⁺ contains a good semisimple element (as defined in§2.1).

(Hk) The residue characteristicpis large enough so that

1) with respect to the adjoint representation G→ GL(g), the expo- nential map (resp. the log map) is defined on the G-domain g₀⁺ of g (resp. on G0⁺), and exp(gx,r) =Gx,r (resp. log(Gx,r) = gx,r) for any x∈ B(G, k) andr >0,

2) the hypotheses in [5, 3.5.2] and [4, 4.4] are valid.

Groups satisfying (HB) and (HGT) are discussed in [3]. From Proposition 4.1 and Proposition 5.4 of [3], we see that if pis large enough, (HB) and (HGT) are valid. For (Hk), you can find more precise bounds using the Campbell- Hausdorff formula (see [12, 3.1.1, 3.2.3]). Hypothesis (HB) is included in [5, 3.5.2]. Hypothesis (Hk), 2) is necessary to apply results of [4] and [5]. We refer to [5, 3.5.2] and [4, 4.4] for details.

1.4. Partition of gvia good cosets

Definition 1.4.1. — Let S and S⁰ be two good cosets in g. We say that S is associated to S⁰ if there isg ∈G such thatS ∩^gS⁰ 6=∅. We say that S is weakly associated toS⁰ and writeS ∼ S⁰, if eitherS andS⁰ are 0-good cosets, or if d(S) = d(S⁰) = r and there is a sequence of nondegenerate dual cosets S = Y0,Y1,· · ·,Yk = S⁰ of the form Yi = Yi+gxi,r⁺ and Yi ∈ gxi,r, such that Yi∩^giYi+16=∅for somegi∈G.

It is straightforward that the above weak associativity is an equivalence relation. We denote by S the set of equivalence classes of weakly associated good cosets. The following is a corollary of Lemma 1.3.2 and Lemma 1.3.4.

It characterizes weakly associated good cosets.

Lemma 1.4.2. — LetΓbe a good element and G⁰=C_G(Γ). Let S= Γ +gx,r⁺

andS⁰= Γ⁰+gx⁰,r⁺ be two good cosets. IfS ∼ S⁰, then there existg∈Gand y∈ B(G⁰, k)such that ^g(Γ⁰+gx⁰,r⁺)⊂Γ +gy,r⁺.

Proof. — By Lemma 1.3.4, there existh∈Gx,0⁺andn⁰∈g⁰_hx0,r∩ N such that hx⁰ ∈ B(G⁰, k) and^h(Γ⁰+gx⁰,r⁺) = Γ+n⁰+ghx⁰,r⁺. Now, by Lemma 1.3.2, there

o

existh⁰∈G⁰_hx0,0andy∈ B(G⁰, k) such that^h0h(Γ⁰+gx⁰,r⁺) = Γ+^h0n⁰+ghx⁰,r⁺⊂ Γ +gy,r⁺.

As usual, we use the same notation for an equivalence class and its repre- sentative when this will not lead to any confusion.

Definition 1.4.3. — For S ∈S, defineg_S ⊂gas follows:

g_S := [

S⁰∼S GS⁰.

Example 1.4.4. — If S is a 0-good set, thengS =g0. IfS= Γ +gx,r⁺, from Lemma 1.4.2 and Remark 1.3.3, 1), we have gS = ^G(Γ +g⁰_r+). The second example is due to F. Murnaghan.

The following lemma shows that the weak associativity on good cosets in- duces a partition of g, and in turn, a spectral decomposition of the delta dis- tribution ong.

Lemma 1.4.5. — Suppose (HGT)holds. Then, 1) gis the disjoint union ofg_S, S ∈S:

g= [◦ S∈S

g_S.

2) Each g_S is aG-domain, that is, a G-invariant open and closed subset of gin thep-adic topology.

3) For anyf ∈C_c^∞(g), we can decompose R

gf(X)dX as follows:

Z

g

f(X)dX= X

S∈S

Z

g_S

f(X)dX.

Proof. — Part 3) is a consequence of 1) and 2). To prove 1), letX ∈gwith Jordan decomposition Xs+Xn. Let T be a maximal k-torus containing Xs

with d(Xs) =r. If r≥0,X ∈gS where S is a 0-good coset. Ifr < 0, from (HGT), there is a good element Γ in tr\tr⁺ such that Γ ≡ Xs (modgx,r⁺) for any x ∈ B(C_G(Xs), k). Since Xn ∈ CG(Xs) and d(Xn) = ∞, one can find y ∈ B(C_G(Xs), k) such that Xn ∈g_y,r⁺. Then sincey ∈ B(C_G(Xs), k)⊂ B(C_G(Γ), k), we seeX =Xs+Xn is contained in a good coset Γ +g_y,r⁺.

To prove 2), observe thatg_S is obviouslyG-invariant. Eachg_S is open since eachS is open andgS is the union of^GS⁰ withS⁰ ∼ S. It is closed because its complement is a union of open sets from 1).

Remark 1.4.6. — From Remark 1.2.3, ifS ∈Sis not a 0-good coset, we have for anyX∈g_S, d(X) = d(S).If hypothesis (HGT) holds, we also observe that

gS ⊂(g_d(S)\g_d(S)+) = [

S⁰∈S d(S⁰)=d(S)

g_S0.

2. Unrefined minimalK-types and Ge

2.1. Unrefined minimal K-types and dual blobs. — Here, we recall some results from [14], [15], and define dual blobs. For the purpose of simpler explanation, as in [14], we assume that there is a natural isomorphism

φ:Gx,r/Gx,r⁺−→gx,r/gx,r⁺

whenr >0. By [20, 2.4], such an isomorphism exists wheneverGsplits over a tamely ramified extension of k(see also [1, 1.6]).

Definition 2.1.1 (see [14], [15]). — An unrefined minimal K-type (or mini- mal K-type) is a pair (Gx,%, χ), where x ∈ B(G, k), % is a nonnegative real number,χis a representation ofGx,%trivial onGx,%⁺ and

(i) if %= 0,χis a cuspidal representation ofGx,0/Gx,0⁺ inflated toGx,0; (ii) if % >0,χis a nondegenerate character of Gx,%/Gx,%⁺.

The%in the above definition is called thedepthof the minimalK-type (Gx,%, χ).

Definition 2.1.2 (see [13, 2.3.4]). — Lets= (Gx,%, χ) be a minimalK-type.

If% >0, we call the cosetS = Γ^∗+g^∗_x,(−%)+ adual blob ofswhenχis realized byS. That is, for anyg∈Gx,%,

χ(g) = Λ Γ^∗ φ(g) .

If%= 0, we define the dual blob ofsto beg^∗_x,0. We denote bys^]the dual blob ofs.

If % >0, any cosetX^∗+g^∗_x,(−%)+ in g^∗_x,−% defines a character of Gx,%. We denote byχ_X∗ the character ofGx,%represented by X^∗+g^∗_x,(−%)+ when there is no confusion.

Recall that a coset X^∗+g^∗_x,(−%)+ in g^∗ is nondegenerate if X^∗+g^∗_x,(−%)+

does not contain any nilpotent element. A characterχofGx,%is nondegenerate if the dual blob of (Gx,%, χ) is nondegenerate.

Definition 2.1.3 (see [14]). — Two minimalK-typess= (Gx,r, χ) and s⁰ = (Gy,s, ξ) are said to beassociates if they have the same depth, and

1) if r = 0, there is a g ∈ G such that Gx,0∩Ggy,0 surjects onto both Gx,0/Gx,0⁺ andGgy,0/Ggy,0⁺, andχis isomorphic toξ^g;

2) if r >0, theG-orbit of the dual blobs^] intersects withs^0].

o

Theorem 2.1.4 (see [15, 3.5]). — Given (π, Vπ)∈ G, there is a nonnegativee rational number %(π) with the following properties.

1) For some x ∈ B(G, k), the space Vπ^Gx,%(π)+ of Gx,%(π)⁺-fixed vectors is nonzero and %(π)is the smallest number with this property.

2) For anyy∈ B(G, k), if W =Vπ^Gy,%(π)+ 6={0}, then

(i) if %(π) = 0, any irreducible Gy,%(π)-submodule of W contains a minimal K-type of depth zero of a parahoric Gz,0⊂Gy,0;

(ii) if %(π)>0, any irreducible Gy,%(π)-submodule of W is a minimal K-type.

Moreover, any two minimalK-types contained inπ are associates of each other.

The rational number%(π) in the theorem is called thedepthof the irreducible admissible representation (π, Vπ).

2.2. Partition of Ge via minimalK-types

Definition 2.2.1. — Assume the hypothesis (HB) holds. Letsands⁰ be two minimal K-types of depth %. We say sis weakly associated to s⁰ if there is a sequence of minimal K-types of s= s0,s1, . . . ,sk = s⁰ depth % such that the dual blobs s^]_i and s^]_i+1 of si and si+1 satisfy that s^]_i ∩^gis^]_i+1 6= ∅ for some gi∈G.

Recall we defined the dual blob of a minimal K-type (Gx,0, σ) to be gx,0. Hence any two types of depth zero are weakly associated, but they are not necessarily associated. It is straightforward that the above weak associativity is an equivalence relation. On the other hand, associativity is not an equivalence relation.

We denote bySKthe set of equivalence classes of weakly associated minimal K-types.

Remark 2.2.2. — Let s be a minimalK-type, and let s∈SK be its equiva- lence class. Let Gesbe the set of (π, Vπ)∈Ge which contain a minimalK-type weakly associated tos. Note that for anys⁰ ∈s, we haveGes=Ges⁰. Then from Theorem 2.1.4, we have

Ge= [◦ s∈SK

Ges.

Note that ifsis a minimalK-type of depth zero,Gesis the set of all depth zero irreducible representations.

From now on, we use the same notation for an equivalence class and its representative in cases where there is no confusion.

3. Plancherel distributions and good minimalK-types

In this section, we investigate some relations between partitions ofgandG.e These partitions are discussed in the previous sections. As stated in Theo- rem 3.3.1, they give rise to a matching between spectral decomposition factors.

We first review Plancherel formulas.

3.1. Plancherel formulas ongandG. — For (π, Vπ)∈Gbandf ∈C_c^∞(G), we define a function onGb as

fb(π) := Tr π(f)

= Θπ(f).

Then Harish-Chandra’s Plancherel formula (see [7]) states that there is a Borel measure dπ supported onGbsuch that

(3) f(1) =

Z

Gb

bf(π)dπ.

On the other hand, regardinggas a topological group with respect to addi- tion, we have the following isomorphisms:

b

g'g^∗'g.

The first isomorphism is from Pontrjagin duality, and we also have the second isomorphism via the additive character Λ and an appropriate bilinear form ong. Then the Plancherel formula ongcan be formulated as follows: there is an appropriate measure on bgsuch that forf ∈C_c^∞(g),

f(0) = Z

b g

fb(χ)dχ,

where fb ∈ C_c^∞(bg) is the Fourier transformation of f given by fb(χ) = R

b

gf(Y)χ(Y)dY. When (HB) holds, from the identification gb ' g, we can now write

(4) f(0) =

Z

g

fb(X)dX

wherefb(X) =R

gf(Y)Λ(B(X, Y))dY. In (4), theG-invariant measure dXong should satisfy volg(gx,r) volg(g_x,(−r)⁺) = 1 for allx∈ B(G, k) andr∈R.

To relate Plancherel formulas on gandG, letf ∈C_c^∞(g) be supported in a sufficiently small neighborhood of 0. Thenf◦log defines a function inC_c^∞(G).

Combining the Plancherel formulas (3) and (4), we have (Pl)

Z

g

fb(X)dX =f(0) = Z

Gb

Θπ(f◦log)dπ.

o

3.2. Good cosets, good minimal K-types and Ge

Definition 3.2.1 (see [3], [13]). — Assume (HB) is valid. Lets= (Gx,%, χ) be a minimal K-type. We say (Gx,%, χ) is good, if its dual blob s^] is a good coset.

Note that all minimal K-types of depth zero are good. For later use, we record the following corollary of Lemma 1.3.2:

Corollary 3.2.2. — We keep the notation from Lemma1.3.2. Assume(HB) is valid. Suppose % := −r > 0. If (π, Vπ) ∈ Ge contains a minimal K-type (Gx,%, χ_Γ+X0) with dual blob Γ +X⁰+g_x,r⁺, it also contains a good minimal K-type(Gy,%, χΓ) with dual blobΓ +g_y,r⁺.

Definition 3.2.3. — Assume (HB) holds. Let S ∈S and let s= (Gx,%, χ) be a minimalK-type.

1) If s^]∼ S, we say thatsisweakly associated toS.

2) Define the subset GeS of Ge to be the set of all (π, Vπ) ∈ Ge such that (π, Vπ) contains a good minimalK-typeswiths^]∼ S. We also define

GbS :=GeS∩G.b

Remarks 3.2.4. — 1) If S is a 0-good coset, GeS is the set of depth zero representations. Otherwise, the good minimal K-types contained in the repre- sentations inGeS are weakly associated to each other. In particular, the depth of any representation in GeS is−d(S).

2) If sis a goodK-type weakly associated toS, we have GeS ⊂Ges. This is a direct consequence of the Definition 2.2.1. We observe

GeS =

(π, Vπ)∈Ges|(π, Vπ) contains a good minimalK-type . 3) Observe that if S ∼ S⁰, then GeS =GeS⁰. Hence, forS ∈S, we can write GeS without any confusion.

Remark 3.2.5. — Although we will later give another proof, we recall from Theorem 2.4.10 of [13] that any (π, Vπ)∈Ge contains a goodK-type. Combin- ing this with Theorem 2.1.4 implies the following: Suppose (HB) and (HGT) are valid. Let S1,S2∈S and (π, Vπ)∈G.e

(i) If (π, Vπ) contains minimalK-typess1,s2 withsi∼ Si,i= 1,2, then we haveS1∼ S2.

(ii) TheGe is a disjoint union ofGeS’s withS ∈S: Ge= S◦

S∈SGeS. Lemma 3.2.6. — Suppose (HB) is valid.

1) For S ∈S, GbS is open and closed inG.b

2) For f ∈C_c^∞(G), we have Z

Gb

bf(π)dπ= Z

G\b ( S◦

S∈SGbS)

bf(π)dπ+ X

S∈S

Z

GbS

bf(π)dπ.

3) Each GbS has a finite Plancherel volume, that is,vol_Gb(GbS)<∞.

Proof. — 1) Assume d(S)<0 first. We fix a good element Γ such that S = Γ +g_x,(−%)⁺withx∈ B(C_G(Γ), k). ForS⁰∼ S, letsS⁰ = (JS⁰, χS⁰) be the good minimal K-type with dual blob S⁰. Let ∆ be the set of G-conjugacy classes ofsS⁰ withS⁰∼ S. Let

G(Sb ⁰) :=

(π, Vπ)∈Gb|m(χS⁰, Vπ)>0 ,

where m(χS⁰, Vπ) is the multiplicity of χS⁰ in (π, Vπ) ∈ G. Note that (i)b each G(Sb ⁰) is open and closed because m(χS⁰, Vπ) is semicontinuous on G,b (ii) if S⁰ and S⁰⁰ are G-conjugates, G(Sb ⁰) = G(Sb ⁰⁰), (iii) ∆ is finite, and (iv) GbS =S

S⁰∈∆G(Sb ⁰). HenceGbS is open and closed when d(S) <0. If S is a 0-good coset, we have

GbS =

(π, Vπ)∈Gb|Vπ^Gx,0+ 6= 0, for some x∈ B(G, k) . Then this case can be proved in a similar way.

2) This is a consequence of 1).

3) More generally, it is enough to show that for any open compact subgroup J ⊂G, the set Gb^J defined by {(π, Vπ)∈Gb |V_π^J 6= 0}has a finite Plancherel volume. It follows from the following sequence of inequalities: for f ∈C_c^∞(G) given by the characteristic function ofJdivided by volG(J), from the Plancherel formula onG, we have

vol_Gb(Gb^J)≤ Z

Gb^J

dimC(V_π^J)dπ= Z

Gb

dimC(V_π^J)dπ= Z

Gb

bf(π)dπ

=f(1) = 1 volG(J)·

3.3. Main Theorem. — The following is a refinement of the equality (Pl).

That is, we find an equality between summands of the left and right hand sides (see Lemma 1.4.5 and Lemma 3.2.6) of the equation. The proof will be given in the following section.

Theorem 3.3.1. — Suppose hypothesis (Hk) is valid. Let S be a good coset and f ∈C_c^∞(g). If S is a 0-good coset, assume f ∈C_c^∞(g0⁺). If S is a good coset of depth −% <0, assumef ∈C_c^∞(g%). Then, we have

Z

g_S

fb(X)dX = Z

GbS

f\◦log(π)dπ.

o

4. Proof and applications of Theorem 3.3.1

In this section, we prove Theorem 3.3.1. We treat positive depth cases and depth zero cases separately. Before each case, we recall some relevant results from [13] and from [4], [5] respectively.

Recall a distribution T onC_c^∞(g) is aC-valued linear functional onC_c^∞(g).

The distribution T is called invariant if ^gT = T for g ∈ G, where ^gf(X) = f(^g−1X) and ^gT(f) = T(^g−1f) for f ∈C_c^∞(g). We denote the space of invari- ant distributions by J(g). For any subset L ⊂g, we denote the subspace of invariant distributions supported on the closure of ^GL by J(L). For r ∈ R, we define someG-invariant subspaces ofC_c^∞(g) as follows:

Dr:= X

x∈B(G,k)

Cc(g/gx,r), Dr⁺ := X

x∈B(G,k)

Cc(g/gx,r⁺)

For any T ∈ J(g) and D ⊂ C_c^∞(g), we denote the restriction of T to D by resDT. For anyJ ⊂ J(g), let resDJ denote{resDT|T∈ J }.

4.1. Some results on character expansions: positive depth cases Recall from [13, 3.1.2] that we defined some subspaces ofJ(g) to characterize the character distributions of thoseπ which contain a given good minimal K- type.

Definition 4.1.1 (see [13, 3.1.2]). — Let Γ be a good element of depthr <0 (note that Γ 6= 0). Let G⁰ := C_G(Γ) be the centralizer of Γ in G, and let N⁰ :=N ∩g⁰, the set of nilpotent elements ing⁰. We define some subspaces ofJ(g):

1) Forx∈ B(G, k) andr∈R, we define J_x,s,r^Γ + as follows:

• Ifs < r, J_x,s,r^Γ + :=

(

T∈ J(g)

forf ∈C(gx,s/gx,r⁺)

if Supp(f)∩(N +gx,s⁺) =∅ then T(f) = 0

) .

• Ifs=r, J_x,r,r^Γ +:=

(

T∈ J(g)

forf ∈C(gx,r/g_x,r⁺) if Supp(f)∩^G(Γ +N⁰) =∅ then T(f) = 0

) .

2) J_r^Γ+:= \

x∈B(G,k)

\

s≤r

J_x,s,r^Γ +.

ForX ∈g, letO(X) be the set of allG-orbits whose closures containX. For X ∈g⁰, O⁰(X) denotes the set of allG⁰-orbits whose closures ing⁰ containX.

Recall that there is a bijection betweenO⁰(0) and O(Γ) given byn⁰ ↔Γ +n⁰ (see [8]).

For eachO ∈ O(Γ), let µO be the orbital integral associated toO. Denote the span ofµO withO ∈ O(Γ) byJΓ.

Proposition 4.1.2. — Suppose (Hk)holds. LetΓbe a good element of depth r <0, and let G⁰:=C_G(Γ). Then we have

1) resD_r+J_r^Γ+= resD_r+JΓ;

2) for T1,T2∈ J_r^Γ+, resD_r+T1= resD_r+T2 if and only if T1([Γ +v+g_x,r⁺]) = T2([Γ +v+g_x,r⁺]) for each x∈ B(G⁰, k)andv∈ N⁰∩gx,r.

Recall that [Γ +v+gx,r⁺] is the characteristic function supported on the set Γ +v+gx,r⁺.

Proof. — Part 1) follows from [13, 3.1.7]. Part 2) is a corollary of 3.1.7 and 4.2.1 in [13].

In fact, 2) is a weaker statement of the results in [13]. However, 2) is enough for our purposes, and we can avoid introducing new notation included in the stronger statement.

Theorem 4.1.3 (see [13, 5.2.1, 5.3.1]). — Assume that(Hk)is valid. Suppose that (π, Vπ) is an irreducible admissible representation of G of positive depth

%(π) =%. Suppose(π, Vπ)contains a good minimalK-types= (Gx0,%, χΓ). Let Γ +gx0,(−%)⁺ be the dual blob ofscontaining a good element Γ. Then,

1) Θbπ∈ J_(−%)^(−Γ)+,

2) ΘπisΓ-asymptotic ong%. That is, there arecO(π)∈Cindexed byO(Γ), such that for any regular element X ∈g%,

Θπ exp(X)

= X

O∈O(Γ)

cO(π)·µcO(X).

4.2. Proof of Theorem 3.3.1: positive depth cases. — Fix% >0. We keep the notation from the previous section. Let T¹_S and T²_S be invariant distributions defined as follows: for f ∈ C_c^∞(g), denote by f0⁺ the function f ·[g0⁺] where [g0⁺] is the characteristic function supported ong0⁺. Then

T¹_S(f) :=

Z

g_S

df0⁺(X)dX,

T²_S(f) :=

Z

GbS

f0\⁺◦log(π)dπ= Z

GbS

Θπ(f0⁺◦log)dπ.

Then to prove the theorem, it is enough to prove that T¹_S(f) = T²_S(f) for anyf ∈C_c^∞(g%) when d(S) =−%.

o

On the other hand, the Fourier transform mapsC_c^∞(g%) toD(−%)⁺ (see [2, 4.2.3]). We will prove Theorem 3.3.1 by showing that Tb¹_S =Tb²_S onD_(−%)⁺ in this case.

For the rest of this section, we fix a good element Γ such that S= Γ +gx0,(−%)⁺

withx0∈ B(C_G(Γ), k). Let G⁰:=C_G(Γ).

Lemma 4.2.1. — 1) The invariant distributions Tb¹_S and Tb²_S are elements of J_(−%)^(−Γ)+.

2) The restrictions resD_(−%)+Tb¹_S and resD_(−%)+Tb²_S are in the linear span of orbital integrals µO, O ∈ O(Γ).

Proof. — Note that 2) is a consequence of 1) and Proposition 4.1.2-(1).

To prove 1), we see by Theorem 4.1.3, 2), we have Tb²_S ∈ J_(−%)^(−Γ)+. To show Tb¹_S ∈ J_(−%)^(−Γ)+, letx∈ B(G, k) ands≤(−%). We have forf ∈C(gx,s/gx,(−%)⁺),

Tb¹_S(f) = Z

g_S

f(−X)dX.

Then we observe that (i) Supp(Tb¹_S) =−g_S;

(ii) ifs <−%, Supp(bT¹_S) =−g_S ⊂g(−%)⊂ N+g_x,s⁺, by [2, 3.3.2] . If s = (−%), it follows from (i) and the equality gS =^G(Γ +g⁰_(−%)+) that Tb¹_S is in J_{x,−%,(−%)}^(−Γ) +. If s < (−%), Tb¹_S ∈ J_x,s,(−%)^(−Γ) + is a result of (ii). Hence Tb¹_S ∈ J_(−%)^(−Γ)+.

To finish the proof of Theorem 3.3.1, from Proposition 4.1.2, 2), it is enough to check that forx∈ B(G⁰, k) andv∈ N⁰∩gx,r,

(E) Tb¹_S [−Γ +v+g_x,(−%)⁺]

=Tb²_S [−Γ +v+g_x,(−%)⁺] .

Since Supp([−Γ +v+gx,(−%)⁺])⊂g_S, we have (a) Tb¹_S [−Γ +v+gx,(−%)⁺]

= volg(gx,(−%)⁺) = 1 volg(gx,%)·

To compute Tb²_S([−Γ + v + gx,(−%)⁺]), we find the Fourier transform of [−Γ +v+gx,(−%)⁺]. In the following,f denotes [−Γ +v+gx,(−%)⁺].

fb(Y) = Z

g

f(X)Λ B(X, Y) dX=

Z

g

[gx,(−%)⁺](X)·Λ B(−Γ +v+X, Y) dX

= volg(g_x,(−%)⁺)·Λ B(−Γ +v, Y)

·[gx,%](Y)

= 1

volG(Gx,%)·Λ B(−Γ +v, Y)

·[gx,%](Y).

Note that volG(Gx,%) · (fb ◦ log) is a character of Gx,% with dual blob

−Γ +v+gx,(−%)⁺, that is, volG(Gx,%)·(bf◦log) =χ_−Γ+v. Then Θπ(bf◦log) = Θπ(χ_−Γ+v) is the multiplicity m(χ_Γ−v, Vπ) of χ_Γ−v in Vπ. Then by Corol- lary 3.2.2, any (π, Vπ) with m(χ_Γ−v, Vπ) > 0 is contained in GeS. Hence m(χ_Γ−v, Vπ) = 0 unlessπ∈GbS, and we have

(b) 1

volg(gx,%) =fb log(1)

=R

GbΘπ(bf ◦log)dπ

=R

GbSΘπ(bf ◦log)dπ=Tb²_S [−Γ +v+gx,(−%)⁺] .

Now equality (E) follows from (a) and (b). Hence Theorem 3.3.1 is proved when% >0.

4.3. Some results on character expansions: depth zero cases. — The following subspaces of J(g) are defined in [5, 2.1.1, 2.1.3]. Ifr≤0, they char- acterize the character distributions of irreducible admissible representations of depth −r.

Definition 4.3.1. — Letx∈ B(G, k)andr∈R. Fors < r, we defineJ_x,s,r^Γ +

as follows:

J˜x,s,r⁺:=



T∈ J(g)

forf ∈C gx,s/gx,r⁺

if Supp(f)∩(N +g_x,s⁺) =∅ then T(f) = 0



. J˜r⁺:= \

x∈B(G,k)

\

s≤r

J˜x,s,r⁺.

From now on, we fix > 0 such thatg(−)⁺ =g0 and D(−)⁺ =D0. Such anexists. Then we also haveg=g0⁺. WhenGis a classical group,gis the set of topologically nilpotent elements.

The following two propositions are corollaries of the results in [18] and [5].

Although their results are more general, we state only the facts necessary to prove Theorem 3.3.1. Note thatO(0) is the set of nilpotent orbits ing. Denote the span ofµO, O ∈ O(0) byJ0.

o