= Rational extension from SL to GL
{title2=$\rho'(g)|_{W_k}=a^k\rho(a^{-1}g)|_{W_k},\quad a^m=\det g$}
Split a rational $SL_m$ representation by its finite scalar center: on $W_k$, $\rho(\zeta I)=\zeta^kI$ for $\zeta^m=1$. The displayed extension is independent of the scalar-root choice and is a <group homomorphism>. A matrix coefficient of $\rho|_{W_k}$ can be represented on $SL_m$ by a sum of homogeneous <polynomials> $P_d$ with $d\equiv k\pmod m$, by averaging over the scalar center. Its extension is $\sum_d(\det g)^{(k-d)/m}P_d(g)$, a <regular function> on $GL_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.
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