Stabilization of the homotopy groups of the self equivalences of linear spheres

Abstract: Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}ⁿ U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i \geq 0$ the sequence of group $$\dots \rightarrow \pi_i \operatorname{map}^{G(S(V_n),S(V_n))} \rightarrow \pi_i \operatorname{map}^{G(S(V_{n+1}),S(V_{n+1}))} \rightarrow \dots$$ stabilizes with the stable group $\oplus_H \omega_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S(\oplus_n U_n)$.