Parabolic induction of the trivial module as flag functions

The trivial representation $\mathbb{C}$ of $L_{\alpha }$ has $\pi (p)\cdot z=z$ for every $p\in P_{\alpha }$ and $z\in \mathbb{C}$, because $\mathbb{C}$ is the trivial $L_{\alpha }$-module; hence the inflated action of [F1] is $p\cdot z=\pi (p)\cdot z=z$, so the inflation ${\mathrm{Inf}}_{L_{\alpha }}^{P_{\alpha }}\mathbb{C}$ is the trivial $P_{\alpha }$-module.

By [F2] applied to the subgroup $P_{\alpha }\le G$, the induction ${\mathrm{Ind}}_{P_{\alpha }}^G\mathbb{C}$ of the trivial $P_{\alpha }$-module is the permutation representation of $G$ on the left cosets $G\mathrm{/}{P}_{\alpha }$, so by step 1.1 the Harish-Chandra induction satisfies $R_{L_{\alpha }}^G(\mathbb{C})={\mathrm{Ind}}_{P_{\alpha }}^G({\mathrm{Inf}}_{L_{\alpha }}^{P_{\alpha }}\mathbb{C})\cong \mathbb{C}[G\mathrm{/}{P}_{\alpha }]$.

The orbit map $g{P}_{\alpha }\mapsto g\cdot W_{\bullet }^{(\alpha )}$ is a $G$-equivariant bijection $G\mathrm{/}{P}_{\alpha }\to {\mathcal{F}}_{\alpha }$ by [F4]; transporting functions along it defines a linear isomorphism $\mathrm{\Phi }:\mathbb{C}[G\mathrm{/}{P}_{\alpha }]\to \mathbb{C}[{\mathcal{F}}_{\alpha }]$ by $\mathrm{\Phi }(f)(g\cdot W_{\bullet }^{(\alpha )}):=f(g{P}_{\alpha })$, which is well defined and bijective because the orbit map is. It is $G$-equivariant: for $x,g\in G$ one has $\mathrm{\Phi }(x\cdot f)(g\cdot W_{\bullet })=(x\cdot f)(g{P}_{\alpha })=f(x^{-1}g{P}_{\alpha })=\mathrm{\Phi }(f)((x^{-1}g)\cdot W_{\bullet })=\mathrm{\Phi }(f)(x^{-1}\cdot (g\cdot W_{\bullet }))$, using the left-coset action on $G\mathrm{/}{P}_{\alpha }$ and the action of [F4] on flags; hence $\mathbb{C}[G\mathrm{/}{P}_{\alpha }]\cong \mathbb{C}[{\mathcal{F}}_{\alpha }]$ as $G$-modules, with the action $(x\cdot f)(F_{\bullet })=f(x^{-1}\cdot F_{\bullet })$.

Combining steps 2.1 and 3.1 gives $R_{L_{\alpha }}^G(\mathbb{C})\cong \mathbb{C}[{\mathcal{F}}_{\alpha }]$: the Harish-Chandra induction of the trivial $L_{\alpha }$-module is the permutation representation of $G$ on the partial flags of type $\alpha$, the space of complex functions on ${\mathcal{F}}_{\alpha }$ with the action $(x\cdot f)(F_{\bullet })=f(x^{-1}\cdot F_{\bullet })$.

For $\alpha =(n)$ the standard parabolic is $P_{(n)}=G$ with $L_{(n)}=G$ and $U_{(n)}=\{I_n\}$, and by [F4] the set ${\mathcal{F}}_{(n)}$ has the single element $0⊊V$, so $R_G^G(\mathbb{C})$ is the permutation representation on a one-point set, that is, the trivial $G$-module $\mathbb{C}$; for $\alpha =(1^n)$ the identification of [F5] gives $\mathbb{C}[{\mathcal{F}}_{(1^n)}]\cong \mathbb{C}[G\mathrm{/}B]$ with $B=P_{(1^n)}$, the permutation module of $G$ on the complete flags.

Finally ${\dim }_{\mathbb{C}}{R}_{L_{\alpha }}^G(\mathbb{C})=[G:P_{\alpha }]\cdot 1=[G:P_{\alpha }]$ by the dimension formula of [F1], and under the bijection of [F4] the index $[G:P_{\alpha }]$ is the number of partial flags of type $\alpha$; the two extremes of step 5.1 are the cases $\alpha =(n)$, where there is a single type-$\alpha$ flag and the index is $1$, and $\alpha =(1^n)$, where the flags are the complete flags and $R_T^G(\mathbb{C})\cong \mathbb{C}[G\mathrm{/}B]$. ∎