Let $D_k$ denote a discrete series representation of $\text{GL}_2(\mathbb{R})$ of weight $k\geq 2$. Consider the parabolically induced representation $D_k \times D_k$, which is a representation of $\text{GL}_4(\mathbb{R})$. Is the representation $D_k \times D_k$ distinguished with respect to the subgroup $\text{GL}_2(\mathbb{R}) \times \text{GL}_2(\mathbb{R})$ of $\text{GL}_4(\mathbb{R})$? Any help would be greatly appreciated.
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$\begingroup$ When is a representation "distinguished"? $\endgroup$– Kenta SuzukiCommented Oct 16 at 3:35
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$\begingroup$ Distinguished here means that there is a non-zero GL2\times GL2 invariant linear form on the representation space of Dk\times Dk. $\endgroup$– Akash YadavCommented Oct 16 at 6:11
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