Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question]

If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? E.g. I know $$\mathrm{res}^{\mathrm{GL}_n}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}} V(\lambda)_n \cong \bigoplus_{\alpha, \beta} c_{\alpha, \beta}^\lambda V(\alpha)_k \otimes V(\beta)_{n-k}$$ where $V(\lambda)_n$ is the irreducible polynomial representations corresponding to a partition (or Young diagram) $\lambda$ of $\mathrm{GL}_n(\mathbb C)$ and $c^\lambda_{\alpha,\beta}$ are the Littlewood-Richardson numbers. Is it true that $$ \mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n ?$$ (I know it is not but true but it should be true up to being semi-simple.)

Solution 1:

It is not true that $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n.$$ Frobenius reciprocity states that $$\mathrm{Hom}_{\mathrm{GL}_n}( \mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} ,V(\lambda)_n) \cong \mathrm{Hom}_{\mathrm{GL}_k \times \mathrm{GL}_{n-k}}(V(\alpha)_k \otimes V(\beta)_{n-k},\mathrm{res}^{\mathrm{GL}_n}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}} V(\lambda)_n) \cong \mathbb C^{c^\lambda_{\alpha,\beta}}$$ and thus there is a surjection $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \twoheadrightarrow \bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n.$$ More can hardly be said. In particular $\bigoplus_\lambda c_{\alpha,\beta}^\lambda V(\lambda)_n$ is finite dimensional but $$\mathrm{ind}_{\mathrm{GL}_k\times \mathrm{GL}_{n-k}}^{\mathrm{GL}_n} V(\alpha)_k \otimes V(\beta)_{n-k} \cong \bigoplus_{\mathrm{GL_n}/(\mathrm{GL}_k\times \mathrm{GL}_{n-k})} V(\alpha)_k \otimes V(\beta)_{n-k} $$ is infinite dimensional.