Restrictions of unitary representations:
Examples and applications to automorphic forms
Birgit Speh
Cornell University
1
Notation:
G semisimple (reductive) connected Lie group,
g Lie algebra,
K max compact subgroup
g = k p Cartan decomposition with Cartan involution
2
Notation:
G semisimple (reductive) connected Lie group,
g Lie algebra,
K max compact subgroup
g = k p Cartan decomposition with Cartan involution
H reductive subgroup of G with max compact subgroup
KH = H K
h Lie algebra of H
2
Problem:
Suppose that is an irreducible unitary representation of G.
Understand the restriction of to H.
3
Problem:
Suppose that is an irreducible unitary representation of G.
Understand the restriction of to H.
Precisely:
Z
|H = d
M
I want to get information about the discrete part of this integral
decomposition.
3
Problem:
Suppose that is an irreducible unitary representation of G.
Understand the restriction of to H.
Precisely:
Z
|H = d
M
I want to get information about the discrete part of this integral
decomposition.
Applications to the cohomology of discrete groups and automor-
phic forms.
3
Remarks about the restriction of unitary representations.
First case: If as a unitary representation on a Hilbert space
|H = H
then we call
for irreducible unitary representations H H
H-admissible case:
Theorem (Kobayashi)
Suppose that is H-admissible for a symmetric subgroup H.
Then the underlying (g, K) module is a direct sum of irreducible
(h, KH ) -modules ( i.e is infinitesimally H-admissible).
If an irreducible (h, K H) module U is a direct summand of a
H-admissible representation , we say that it is a H-type of .
4
Toshiyuki Kobayashi also obtained sufficient conditions for to
be infinitesimally H-admissible.
Such conditions are subtle as the following example shows:
5
Toshiyuki Kobayashi also obtained sufficient conditions for to
be infinitesimally H-admissible.
Such conditions are subtle as the following example shows:
Example: (joint with B. Orsted)
Let G=SL(4, R). There are 2 conjugacy classes of symplectic
subgroups. Let H1 and H2 be symplectic groups in different
conjugacy classes.
5
Toshiyuki Kobayashi also obtained sufficient conditions for to
be infinitesimally H-admissible.
Such conditions are subtle as the following example shows:
Example: (joint with B. Orsted)
Let G=SL(4, R). There are 2 conjugacy classes of symplectic
subgroups. Let H1 and H2 be symplectic groups in different
conjugacy classes.
There exists an unitary representation of G which is H1 admis-
sible but not H2 admissible .
5
Some representations and their (g, K)-modules
Let gC be the complexification of g and let T be a maximal torus
in K. Then x0 T defines a stable parabolic subalgebra
q C = l C uC
of gC.
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Some representations and their (g, K)-modules
Let gC be the complexification of g and let T be a maximal torus
in K. Then x0 T defines a stable parabolic subalgebra
q C = l C uC
of gC.
For a stable parabolic q and an integral and sufficiently regular
character of q we can construct a family of representations
Aq().
These representations Aq() were constructed by Parthasarathy
using the Dirac operator and also independently using homolog-
ical algebra by G. Zuckerman in 1978. Write Aq := Aq(0)
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Consider G= U(p,q), K=U(p)U(q) with p, q > 1.
H1 =U(p,1) U(q-1)
H3 =U(p-1) U(1,q) .
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Consider G= U(p,q), K=U(p)U(q) with p, q > 1.
H1 =U(p,1) U(q-1)
H3 =U(p-1) U(1,q) .
Suppose q is a -stable parabolic subalgebra defined by x0 U (p).
and Gx0 = L =U(p-r) U(r,q), with p r > 0 .
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Consider G= U(p,q), K=U(p)U(q) with p, q > 1.
H1 =U(p,1) U(q-1)
H3 =U(p-1) U(1,q) .
Suppose q is a -stable parabolic subalgebra defined by x0 U (p).
and Gx0 = L =U(p-r) U(r,q), with p r > 0 .
Theorem 1.
Aq is always H1-admissible
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Consider G= U(p,q), K=U(p)U(q) with p, q > 1.
H1 =U(p,1) U(q-1)
H3 =U(p-1) U(1,q) .
Suppose q is a -stable parabolic subalgebra defined by x0 U (p).
and Gx0 = L =U(p-r) U(r,q), with p r > 0 .
Theorem 1.
Aq is always H1-admissible
If Aq is not holomorphic or anti holomorphic it is not not H3-
admissible.
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A Blattner type formula describes the multiplicities of irreducible
H1-types in the restriction of Aq of U(p,q) to H1 confirming a
conjecture by S-Orsted.
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A Blattner type formula describes the multiplicities of irreducible
H1-types in the restriction of Aq of U(p,q) to H1 confirming a
conjecture by S-Orsted.
**********************
Similar results are also true for the connect component of
G =S0(p,q), H1 connected component of SO(p,q-1) and H3
connected component of SO(p-1,q).
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Applications to the cohomology of discrete groups
G(Q) a torsion-free congruence subgroup.
S() := \X = \G/K is a locally symmetric space.
S() has finite volume under a G-invariant volume form inherited
from X.
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Applications to the cohomology of discrete groups
G(Q) a torsion-free congruence subgroup.
S() := \X = \G/K is a locally symmetric space.
S() has finite volume under a G-invariant volume form inherited
from X.
S() is orientable if is small enough. Fix an orientation.
9
Applications to the cohomology of discrete groups
G(Q) a torsion-free congruence subgroup.
S() := \X = \G/K is a locally symmetric space.
S() has finite volume under a G-invariant volume form inherited
from X.
S() is orientable if is small enough. Fix an orientation.
Consider
H (, C) = HdeRahm
(S(), C).
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By (Matsushima-Murakami)
H (\X, C) = H (g, K, C (\G)).
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By (Matsushima-Murakami)
H (\X, C) = H (g, K, C (\G)).
For an irreducible finite dimensional rep. E of G,
H (\X, E)
= H (g, K, C (\G) E).
10
By (Matsushima-Murakami)
H (\X, C) = H (g, K, C (\G)).
For an irreducible finite dimensional rep. E of G,
H (\X, E)
= H (g, K, C (\G) E).
If be a representation of G we can also define H (g, K, E).
For an irreducible unitary representation
H (g, K, E) = HomK (p, E).
Here g = k p is the Cartan deposition of g.
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Example: If G = U (p, q) then H (g, K, Aq) 6= 0.
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Example: If G = U (p, q) then H (g, K, Aq) 6= 0.
If cocompact L2(\G) = m(, ), and
H (g, K, C (\G)) = G
m(, )H (g, K, ).
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Example: If G = U (p, q) then H (g, K, Aq) 6= 0.
If cocompact L2(\G) = m(, ), and
H (g, K, C (\G)) = G
m(, )H (g, K, ).
Vanishing theorems for H (\X, E) by G. Zuckerman in 1978
and later by Vogan-Zuckerman 1982, nonvanishing theorems by
Li using representation theory and classification of irreducible
representations with nontrivial (g, K)-cohomology.
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Back to the example G = U (p, q) and the representation Aq.
Proposition
If is a H1-type of Aq then there exists a finite dimensional
representation F of H1 so that
H (h1, K H1, F ) 6= 0.
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Back to the example G = U (p, q) and the representation Aq.
Proposition
If is a H1-type of Aq then there exists a finite dimensional
representation F of H1 so that
H (h1, K H1, F ) 6= 0.
Write q = l u and s = dim u p. Then s is the smallest degree
so that
H s(g, K, Aq) = HomK (sp, Aq) 6= 0.
12
Back to the example G = U (p, q) and the representation Aq.
Proposition
If is a H1-type of Aq then there exists a finite dimensional
representation F of H1 so that
H (h1, K H1, F ) 6= 0.
Write q = l u and s = dim u p. Then s is the smallest degree
so that
H s(g, K, Aq) = HomK (sp, Aq) 6= 0.
Let g = h q and q1 = p q .
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Let AqH the H1 type of Aq generated by the minimal K-type of
1
Aq and s1 = dim u p h1 .
There is canonical identification of
HomK (sp, Aq)
and
HomKH1 (s1 (p h1), AqH ss1 q1)
1
Theorem 2.
H s1 (h, K H, AqH ss1 q) 6= 0
1
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Let AqH the H1 type of Aq generated by the minimal K-type of
1
Aq and s1 = dim u p h1 .
There is canonical identification of
HomK (sp, Aq)
and
HomKH1 (s1 (p h1), AqH ss1 q1)
1
Theorem 2.
H s1 (h, K H, AqH ss1 q) 6= 0
1
Remark 1: Under our assumptions: s1 < s.
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Remark 2:
This result combined with Matsushima Murakami and Oda re-
striction of differential forms allows an maps from cohomology
of X\ to a locally symmetric space for H1..
14
Remark 2:
This result combined with Matsushima Murakami and Oda re-
striction of differential forms allows an maps from cohomology
of X\ to a locally symmetric space for H1..
Remark 3:
I conjecture that in the restriction of Aq to H3 there is always a
direct summand AqH whose lowest nontrivial cohomology class
3
is in degree s. Special case of this conjecture was proved by Li
and Harris.
14
More about restrictions of representations, which are not
H-admissible.
15
More about restrictions of representations, which are not
H-admissible.
Theorem 3. (Kobayashi)
Let be an irreducible unitary representations of G and suppose
that U is an irreducible direct summand of . If the intersection
of the underlying (h, K H)module of U with the underlying
(g, K)module of is nontrivial then the representation is H
admissible.
15
More about restrictions of representations, which are not
H-admissible.
Theorem 3. (Kobayashi)
Let be an irreducible unitary representations of G and suppose
that U is an irreducible direct summand of . If the intersection
of the underlying (h, K H)module of U with the underlying
(g, K)module of is nontrivial then the representation is H
admissible.
Consider G=SO(n,1), L= SO(2r) SO(n-2r,1) , 2r 6= n and
H= SO(n-1,1) SO(1). The representation Aq is not H-admissible,
so Kobayashis theorem implies that finding direct summands is
an analysis problem and not an algebra problem.
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Warning: There exists a unitary representation of SL(2,C)
whose restriction to SL(2,R) contains a direct summand but
doesnt contain any smooth vectors of . (joint with Venkatara-
mana)
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Warning: There exists a unitary representation of SL(2,C)
whose restriction to SL(2,R) contains a direct summand but
doesnt contain any smooth vectors of . (joint with Venkatara-
mana)
nonspherical principal series representation of SL(2,C) with
infinitesimal character . The restriction to SL(2,R) has the
discrete series D+ D with infinitesimal character H as direct
summand, but
(D+ D)
Proof uses concrete models of the representations. J. Vargas
recently proved more general case.
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Restriction of complementary series representations.
Let G = SL(2, C), B(C) the Borel subgroup of upper triangular
matrices in G,
and
!
a n 2.
( ) =| a |
0 a1
For u C
u = {f C (G)| f (bg) = (b)1+uf (g)
for all b B(C) and all g G(C) and in addition are SU (2)-finite.
For 0 < u < 1 completion to the unitary complementary series
rep
u with respect to an inner product < , >u .
17
Similiar define the complementary series
t of H=Sl(2,R) for
0 < t < 1.
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Similiar define the complementary series
t of H=Sl(2,R) for
0 < t < 1.
Theorem 4. (Mucunda 74) Let 1 2 < u < 1 and t = 2u 1. The
complementary series representation b t of SL(2, R) is a direct
summand of the restriction of the complementary series repre-
sentation
b u of SL(2, C).
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Similiar define the complementary series
t of H=Sl(2,R) for
0 < t < 1.
Theorem 4. (Mucunda 74) Let 1 2 < u < 1 and t = 2u 1. The
complementary series representation b t of SL(2, R) is a direct
summand of the restriction of the complementary series repre-
sentation
b u of SL(2, C).
Idea of a different proof jointly with Venkataramana: Consider
the geometric restriction res of functions on G to functions on
H. We show that
res : u t
is continuous with respect to the inner products < , >u and
< , > t
18
More precisely we prove
Theorem 5. (joint with Venkataramana)
There exists a constant C such that for all u, the estimate
C || ||2
u || res() ||2
(2u1) .
holds.
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More precisely we prove
Theorem 5. (joint with Venkataramana)
There exists a constant C such that for all u, the estimate
C || ||2
u || res() ||2
(2u1) .
holds.
We conjecture that similar estimates hold for the geometric re-
striction map of groups G of rank one of the subgroups H of the
same type.
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Generalization to the restriction of complementary series
representations of G=SO(n,1) to H=SO(n-1,1).
n standard representation of SO(n-2).
2i we have a unitary complementary series
For 0 < s < 1 n1
representation
R(n, in, s) = IndG i
P n
1s
with the (g, K)modules
r(n, in, s) = indG
P i 1s
n
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Theorem 6. (joint with Venkataramana)
If
1 2i
<s< and i [n/2] 1,
n1 ni
then
i (n 1)s 1
R(n 1, n1, )
n2
occurs discretely in the restriction of the complementary series
representation R(n, in, s) to SO(n-1,1).
21
Theorem 6. (joint with Venkataramana)
If
1 2i
<s< and i [n/2] 1,
n1 ni
then
i (n 1)s 1
R(n 1, n1, )
n2
occurs discretely in the restriction of the complementary series
representation R(n, in, s) to SO(n-1,1).
2i the representation R(n, i , s) tends
As s tends to the limit ni n
to a representation Anqi in the Fell topology.
21
Theorem 7. (joint with Venkataramana)
The representation An1qi of SO(n-1,1) occurs discretely in the
restriction of the representation An
qi of SO(n,1).
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Theorem 7. (joint with Venkataramana)
The representation An1qi of SO(n-1,1) occurs discretely in the
restriction of the representation An
qi of SO(n,1).
Applications to Automorphic forms
The representation An
qi of SO(n,1) is the unique representation
of SO(n,1) with nontrivial (g, K)- cohomology in degree i.
It is tempered for i=[n/2].
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The representation An
qi is not isolated in the unitary dual SO(n,1).
23
The representation An
qi is not isolated in the unitary dual SO(n,1).
The automorphic dual of G is the set of all unitary represen-
tations which isomorphic to a representation in L2(G/) for an
arithmetic subgroup .
23
The representation An
qi is not isolated in the unitary dual SO(n,1).
The automorphic dual of G is the set of all unitary represen-
tations which isomorphic to a representation in L2(G/) for an
arithmetic subgroup .
Theorem 8. (joint with Venkataramana)
If for all n, the tempered representation An qi (i.e. when i =
[n/2]) is not a limit of complementary series in the automorphic
dual of SO(n, 1), then for all integers n, and for i < [n/2], the
cohomological representation An qi is isolated (in the Fell topology)
in the automorphic dual of SO(n, 1).
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Conjecture (Bergeron)
Let X be the real hyperbolic n-space and SO(n, 1) a con-
gruence arithmetic subgroup. Then non-zero eigenvalues of
the Laplacian acting on the space i(X) of differential forms of
degree i satisfy:
>
for some > 0 independent of the congruence subgroup , pro-
vided i is strictly less than the middle dimension (i.e. i [n/2]).
24
Conjecture (Bergeron)
Let X be the real hyperbolic n-space and SO(n, 1) a con-
gruence arithmetic subgroup. Then non-zero eigenvalues of
the Laplacian acting on the space i(X) of differential forms of
degree i satisfy:
>
for some > 0 independent of the congruence subgroup , pro-
vided i is strictly less than the middle dimension (i.e. i [n/2]).
Evidence for this conjecture
For n=2 Selberg proved that Eigen values of the Laplacian
on function satisfy > 3/16 and more generally Clozel showed
there exists a lower bound on the eigenvalues of the Laplacian
on functions independent of .
24
A consequence of the previous theorem:
Corrollary(Joint with Venkataramana)
If the above conjecture holds true in the middle degree for all
even integers n, then the conjecture holds for arbitrary degrees
of the differential forms
25
Happy Birthday, Gregg
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