Non-vanishing cohomology classes in uniform lattices of $\text{SO}(n,\mathbb{H})$ and automorphic representations

Parameswaran Sankaran, Arghya Mondal

Publication date: 5 October 2016

Abstract: Let

X

denote the non-compact globally Hermitian symmetric space of type

D I I I

, namely,

e x t S O (n, m a t h b b H) / e x t U (n)

. Let

L a m b d a

be a uniform torsionless lattice in

e x t S O (n, m a t h b b H)

. In this note we construct certain complex analytic submanifolds in the locally symmetric space

for certain finite index sub lattices

G a m m a s u b s e t L a m b d a

and show that their dual cohomology classes in

H^{*} (X_{G} a m m a; m a t h b b C)

are not in the image of the Matsushima homomorphism

H^{*} (X_{u}; m a t h b b C) o H^{*} (X_{G} a m m a; m a t h b b C)

, where

X_{u} = e x t S O (2 n) / e x t U (n)

is the compact dual of

X

. These submanifold arise as sub-locally symmetric spaces which are totally geodesic, and, when

L a m b d a

satisfies certain additional conditions, they are non-vanishing `special cycles'. Using the fact that

X_{L} a m b d a

is a K"ahler manifold, we deduce the occurrence in

of a certain irreducible representation

(m a t h c a l A_{m} a t h f r a k q, A_{m} a t h f r a k q)

with non-zero multiplicity when

n g e 9

. The representation

m a t h c a l A_{m} a t h f r a k q

is associated to a certain

h e t a

-stable parabolic subalgebra

m a t h f r a k q

of

m a t h f r a k g_{0} : = m a t h f r a k s o (n, m a t h b b H)

. Denoting the smooth

e x t U (n)

-finite vectors of

A_{m a t h f r a k q}

by

A_{m a t h f r a k q, e x t U (n)}

, the representation

m a t h c a l A_{m} a t h f r a k q

is characterised by the property that

H^{p, p} (m a t h f r a k g_{0} o t i m e s m a t h b b C, e x t U (n); A_{m a t h f r a k q, e x t U (n)}) c o n g H^{p - n + 2, p - n + 2} (e x t S O (2 n - 2) / e x t U (n - 1); m a t h b b C), p g e 0

, for

n g e 9

.

Mathematics Subject Classification ID

Semisimple Lie groups and their representations (22E46) Discrete subgroups of Lie groups (22E40)

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