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Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary representations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have all vanishing odd degree Betti numbers? I am interested in some examples and some necessary conditions.

Example: Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$, see "THE SIEGEL MODULAR VARIETY OF DEGREE TWO AND LEVEL THREE" by J. WILLIAM HOFFMAN and STEVEN H. WEINTRAUB.

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary representations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have all vanishing odd degree Betti numbers? I am interested in some examples and some necessary conditions.

Example: Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$, see "The Siegel modular variety of degree two and level three" by J. William Hoffman and Steven H. Weintraub (Trans AMS 2001).

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary representations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have all vanishing odd degree Betti numbers? I am interested in some examples and some necessary conditions.

Example: Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$.

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary representations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have all vanishing odd degree Betti numbers? I am interested in some examples and some necessary conditions.

Example: Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$, see "THE SIEGEL MODULAR VARIETY OF DEGREE TWO AND LEVEL THREE" by J. WILLIAM HOFFMAN and STEVEN H. WEINTRAUB.

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary represetations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have no odd degree Betti numbers? I am interested in some examples and some necessary conditions.

"Example": Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$.

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\Gamma \backslash G/K$ can be expressed in term of unitary representations of $G$ by Matsushima's formula.

The formula looks quite complicated in general, the question is: when does $X_{\Gamma}$ have all vanishing odd degree Betti numbers? I am interested in some examples and some necessary conditions.

Example: Consider the toroidal compactification of Siegel threefold with $3$ level structure, its Betti numbers are $1, 0, 61, 0, 61, 0, 1$.