Uniqueness'1$

Uniqueness of certain mixed models: a new descent method*

Chen-Bo Zhu
Department of Mathematics
National University of Singapore

March 9, 2009
Otsu City, Japan*Joint work with Dihua Jiang and Binyong Sun&%Uniqueness'12$

Main result

Let K = R or C. Let G = GL6 (K) and S be its Ginzburg-Rallis
subgroup:




 a b d 
S= 0 a c 
 | a ∈ GL2 (K) .

0 0 aDefine a character χS of S by 
1 b da 0 0  = χK× (det(a)) ψK (tr(b + c)),χS  0 1 c  ·  0 a 0 
0 0 10 0 a
where ψK is a nontrivial unitary character of K, and χK× is any
character of K× .
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Theorem (uniqueness of Ginzburg-Rallis models)Let V be an irreducible, admissible, smooth Fréchet representationof G of moderate growth. Then we have
dim HomS (V, CχS ) ≤ 1.
Remark: smooth Fréchet representations of moderate growth =(canonical) smooth globalization of Harish-Chandra modules, byCasselman-Wallach.
Terminologies: Whittaker models, linear models, mixed models&%

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Reduction through Gelfand-Kazhdan
criterion

Denote by ∆ the Casimir element (with respect to real trace form),
viewed as a bi-invariant differential operator on G.
Theorem: Let f be a tempered generalized function on G, which
is an eigenvector of ∆. If f satisfies
f (sx) = f (xsτ ) = χS (s)f (x),for all s ∈ S,

thenf (x) = f (xτ ).

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Generalized functions and differentialoperators

Let M be a smooth manifold.
• C−∞ (M ): the space of generalized functions on M .
• For a locally closed subset Z of M , denoteC−∞ (M ; Z) = {f ∈ C−∞ (U )| supp(f ) ⊆ Z},
where U is any open subset containing Z as a closed subset.
• For a differential operator D on M , denoteC−∞ (M ; D) = {f ∈ C−∞ (M )| Df = 0}.
• Given Z and D, denoteC−∞ (M ; Z; D) = C−∞ (M ; Z) ∩ C−∞ (M ; D).
&%Uniqueness'6$Suppose that a Lie group H acts smoothly on a manifold M .
• For a character χ of H, denote C−∞
χ (M ) the space ofχ-equivariant generalized functions.
• If a locally closed subset Z of M is H stable, denote by
−∞
C−∞
(M;Z)
thespaceofallfin
C(M ; Z) which areχχ-equivariant.
−∞• Similar notations: C−∞χ (M ; D); Cχ (M ; Z; D).

If M is a Nash manifold (those which is modeled locally bysemialgebraic sets in Rn ), then one may define a notion oftemperedness.
• C−ξ (M ): the space of tempered generalized functions on M .
• Similar notations such as C−ξχ (M ; Z; D).
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Generalized functions on G

We consider G as a G × G manifold:(g1 , g2 )x = g1 xg2τ .
• For the problem at hand, the relevant subgroup is H = S × S,
and the character of H is χ = χS £ χS .
• We need to show that every generalized function inC−ξχ (G; ∆ − λ) is τ -invariant, for any λ.
• Following Bruhat, we decompose G into P -P τ double cosets:
GG=GRR

where P is the parabolic subgroup of containing S, and try tounderstand such generalized functions by imposing additional
support conditions with respect to double cosets.
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Three ingredients of the proof(a) Reduction to an H stable open submanifold G0 of G consistingof four double cosets, using the transversality of certain
vector fields to all GR ’s outside G0 . The technique is due toShalika.
(b) A descent argument from G0 to a smaller H stable open
submanifold G00 , such that the corresponding problem on G00 is
reduced to the following linear model problems: the uniquenessof trilinear models for GL2 , and the multiplicity one property
for the pair (GL2 , GL1 ). This argument relies on two geometricnotions attached to submanifolds.
(c) Use of the oscillator representation to conclude the uniquenessof the two afore-mentioned linear models. The specificphenomenon is called first occurrence.
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Reduction to G0 by transversality

Lemma: Let D1 be a differential operator on M of order k ≥ 1,which is transversal to a submanifold Z of M . Let D2 be adifferential operator on M which is tangential to Z. ThenC−∞ (M ; Z; D1 + D2 ) = 0.
Using “transversality” outside G0 and the above lemma, we have
Proposition: Let f ∈ C−∞
χ (G). If f is an eigenvector of ∆, and fvanishes on G0 , then f = 0.
Remark: this step cuts off certain small submanifolds of G (thosewhich are outside G0 ).
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First occurrence

For Step (c), we appeal to the following
Lemma: Let E be a finite dimensional non-degenerate quadratic
space over K, and let the orthogonal group O(E) act on E kdiagonally, where k is a positive integer. If k < dim E, and if atempered generalized function f on E k is SO(E)-invariant, then fis O(E)-invariant.
The above lemma may be stated as that the determinant characterof O(E) does not occur in Howe duality correspondence of(O(E), Sp(2k)) if k < dim E. In fact the determinant characteroccurs if and only if k ≥ dim E.
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Some geometric notions

Unipotent χ-incompatibility
Definition: An H stable submanifold Z of M is said to be
unipotently χ-incompatible if for every z0 ∈ Z, there is a local H
slice Z of Z, containing z0 , and a smooth map φ : Z → H such thatthe followings hold for all z ∈ Z:(a) φ(z) ∈ Stabz ,(b) χ(φ(z)) 6= 1, and(c) the linear map
Tz (M )/ Tz (Z) → Tz (M )/ Tz (Z)induced by the action of φ(z) on M is unipotent.
Key Lemma: Let Z be an H stable submanifold of M which isunipotently χ-incompatible. Then C−∞
χ (M ; Z) = 0.
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Metrical properness
Let M is be a pseudo Riemannian manifold.
Definition:(a) A submanifold Z of M is said to be metrically proper if for allz ∈ Z, the tangent space Tz (Z) is contained in a proper
nondegenerate subspace of Tz (M ).
(b) A second order differential operator D is said to be ofLaplacian type if for all x ∈ M , the principal symbol
σ2 (D)(x) = u1 v1 + u2 v2 + · · · + um vm ,where u1 , u2 , · · · , um is a basis of the tangent space Tx (M ),and v1 , v2 , · · · , vm is the dual basis in Tx (M ).
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Note that a Laplacian type differential operator is transversal toany metrically proper submanifold, from its very definition.
Lemma: Let Z be a metrically proper submanifold of M , and let
D be a Laplacian type differential operator on M . ThenC−∞ (M ; Z; D) = 0.
Remark: This is a form of uncertainty principle.

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Uχ M property
Let H be a Lie group acting smoothly on a pseudo Riemannianmanifold M , and let χ be a character on H.
Definition: We say that an H stable locally closed subset Z of M
has Uχ M property if there is a finite filtration
Z = Z0 ⊃ Z1 ⊃ · · · ⊃ Zk ⊃ Zk+1 = ∅of Z by H stable closed subsets of Z such that each Zi \ Zi+1 is asubmanifold of M which is either unipotently χ-incompatible or
metrically proper in M .
Lemma: Let Z be an H stable closed subset of M having Uχ M
property. Then for any differential operator D of Laplacian type,
we haveC−∞χ (M ; Z; D) = 0.
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From G0 to G00 by Uχ M

We shall define an H stable open submanifold G00 of G0 which has
three submanifolds M2 , M3 , M̌3 as local H slices.
Set
M2 = GL2 (K) × GL2 (K) = GL2 (K) × GL2 (K) × {I2 } ⊂ G,
which is stable under the subgroup
H2 = GL2 (K) = {(x, x−τ ) | x ∈ GL∆
2 (K)} ⊂ H.

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Uniqueness'Set∗ ∗ ∗
M3 =  0

 0


 016$

∗

∗0∗

∗

0

∗

x33

000x4400000000



0 0 


0 0 

 ∈ G | x33 6= x44 ,

0 0 








1 0 




0 1

which is stable under a certain (non-reductive) subgroup H3 of H.
Also define a symmetric counterpart of M3 , called M̌3 .

&%Uniqueness

'17$

Let
G00 = HM2 ∪ HM3 ∪ H M̌3 ⊂ G0 .
Proposition: As an H manifold, G0 \ G00 has Uχ M property.
0Consequently if f ∈ C−∞(G) is an eigenvector of ∆, and fχvanishes on G00 , then f = 0.
What we need to do: determine G0 \ G00 explicitly and perform
orbit analysis painstakingly to show the Uχ M property (findingslices and stabilizers to check unipotent χ-incompatibility; finding
tangent spaces to check metrical properness).

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'10

18$

The smaller models or the descents

Proposition: Let H2 = GL2 (K) act on M2 = GL2 (K) × GL2 (K) byg(x, y) = (gxg −1 , gyg −1 ),g ∈ GL2 (K),Then any H2 -invariant tempered generalized function on M2 isτ -invariant, whereτ (x, y) = (xτ , y τ ).
Proof: Extend the action of H2 on M2 to the larger space
gl2 (K) × gl2 (K). It suffices to prove the same on gl2 (K) × gl2 (K),and in fact on sl2 (K) × sl2 (K).
View sl2 (K) as a three-dimensional quadratic space under the trace
form. The action of H2 yields the diagonal action of SO(sl2 (K)) onsl2 (K) × sl2 (K). By first occurrence lemma, any H2 -invariant
tempered generalized function must be O(sl2 (K))-invariant. ¤&%Uniqueness'19$

Remark: the above proposition gives the multiplicity one property
for the pair (GL2 × GL2 , ∆GL2 ), or equivalently the uniqueness oftrilinear models for GL2 .
Now for the H3 action on M3 (and similarly for M̌3 ):• It amounts to uniqueness of certain mixed models for GL3 .
• Further reduced to the multiplicity one property for the pair(GL2 , GL1 ), by using a form of Uχ M property.
• It also follows from first occurrence lemma, now applied to atwo-dimensional quadratic space.

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Final remarks• Apart from the “ultimate” models, the descent process alsopresents other intermediate models such as certain mixed
models for GL4 × GL2 .
• One can prove exactly the same result for G = GL3 (D), where
D is the quaternion division algebra. This is the non-split case.
• One can give a quick proof of the uniqueness of Whittakermodels, by using the notion of unipotent χ-incompatibility.

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