Mackey support of Homs between finite principal series

Harish-Chandra adjunction with $L=T$, $P=B$, $V=\chi$ and $X=I({\chi }^{\mathrm{\prime }})=R_T^G({\chi }^{\mathrm{\prime }})$ gives a $\mathbb{C}$-linear isomorphism ${\mathrm{Hom}}_G(I(\chi ),I({\chi }^{\mathrm{\prime }}))\cong {\mathrm{Hom}}_T(\chi ,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_T^G(I({\chi }^{\mathrm{\prime }})))$.

Specialize the Mackey formula of [F2] to $\alpha =\beta =(1^n)$ and $X={\chi }^{\mathrm{\prime }}$: here $L=M=T$, and for the representative $\dot{w}=P_w$ of a double coset one has $M_w=\dot{w}T{\dot{w}}^{-1}=T$ and $V_w=\dot{w}U{\dot{w}}^{-1}$, so $C_w=T$, $A_w=T\cap V_w=\{1\}$ because unipotent elements are conjugate to unipotent ones and the only diagonal unipotent matrix is the identity, and $D_w=U\cap T=\{1\}$; hence $R_{C_w}^L$ is the identity functor and $({}^w{\chi }^{\mathrm{\prime }}{)}^{D_w}={}^w{\chi }^{\mathrm{\prime }}$. As the double cosets $B\mathrm{\backslash }G\mathrm{/}B$ are indexed by $S_n$ with representatives $\dot{w}$ by [F2], this gives an isomorphism of $T$-modules ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_T^G(R_T^G({\chi }^{\mathrm{\prime }}))\cong {⨁}_{w\in S_n}{}^w{\chi }^{\mathrm{\prime }}\mathrm{.}$

Substituting step 1.2 into step 1.1 and distributing the finite direct sum over ${\mathrm{Hom}}_T(\chi ,-)$ gives ${\mathrm{Hom}}_G(I(\chi ),I({\chi }^{\mathrm{\prime }}))\cong {⨁}_{w\in S_n}{\mathrm{Hom}}_T(\chi ,{}^w{\chi }^{\mathrm{\prime }})$. By [F3] each summand is a $\mathbb{C}$-vector space of dimension $1$ when $\chi =w\cdot {\chi }^{\mathrm{\prime }}$, and dimension $0$ otherwise, because a nonzero homomorphism between the one-dimensional characters $\chi$ and ${}^w{\chi }^{\mathrm{\prime }}$ exists exactly when they are equal; here ${}^w{\chi }^{\mathrm{\prime }}=w\cdot {\chi }^{\mathrm{\prime }}$ by [F2].

Taking dimensions in step 2.1 gives ${\dim }_{\mathbb{C}}{\mathrm{Hom}}_G(I(\chi ),I({\chi }^{\mathrm{\prime }}))=\mathrm{\#}\{w\in S_n:\chi =w\cdot {\chi }^{\mathrm{\prime }}\}$, and this number is nonzero precisely when ${\chi }^{\mathrm{\prime }}$ lies in the $S_n$-orbit of $\chi$, since $w\cdot {\chi }^{\mathrm{\prime }}=\chi$ for some $w$ is exactly the statement that ${\chi }^{\mathrm{\prime }}$ and $\chi$ lie in one orbit. The same decomposition exhibits ${\mathrm{Hom}}_G(I(\chi ),I({\chi }^{\mathrm{\prime }}))$ as the direct sum of the one-dimensional subspaces carried by the indices $w$ with $\chi =w\cdot {\chi }^{\mathrm{\prime }}$; picking any nonzero element in each of these subspaces gives a basis indexed by that set, and any two such choices differ by nonzero scalars.

Steps 1.1 and 1.2 produce the isomorphism, step 2.1 identifies its summands, and step 3.1 records the dimension count, the nonvanishing criterion and the basis statement; all sums are finite over the finite group $S_n$, and every map used is a given adjunction or Mackey isomorphism, so no choice principle is used.