The projective Clifford correspondence for an invariant irreducible representation

Let $U$ be an irreducible complex $I$-module with $U\in \mathrm{Irr}(I\mid \theta )$, so $\theta$ occurs in $U{\mathrm{\mid }}_N$ by [F2]. Since $I_I(\theta )=I$, [F5] shows that $U{\mathrm{\mid }}_N$ is $\theta$-isotypical, so $U$ is a finite-dimensional $\theta$-isotypical $N$-module with $U\mathrm{\ne }0$; put $M:={\mathrm{Hom}}_N(S,U)$ with the trivial $N$-action, a finite-dimensional space that is nonzero because $\theta$ occurs in $U{\mathrm{\mid }}_N$ and $U{\mathrm{\mid }}_N$ is isotypical, so that the multiplicity space of [F6] does not vanish.

For $i\in I$ define $M(i):M\to M$ by $(M(i)f)(s):=U(i)f(P(i{)}^{-1}s)$. This is $\mathbb{C}$-linear and has image in ${\mathrm{Hom}}_N(S,U)$: for $m\in N$, $(M(i)f)(\rho (m)s)=U(i)f(P(i{)}^{-1}\rho (m)s)=U(i)f(\rho (i^{-1}mi)P(i{)}^{-1}s)=U(i)U(i^{-1}mi)f(P(i{)}^{-1}s)=U(m)U(i)f(P(i{)}^{-1}s)=U(m)(M(i)f)(s)$, using $P(i{)}^{-1}\rho (m)=\rho (i^{-1}mi)P(i{)}^{-1}$ from the identities of [F1] and $f\in {\mathrm{Hom}}_N(S,U)$.

The operators $M(i)$ depend only on the coset $iN$ and $M(n)={\mathrm{id}}_M$ for $n\in N$. Indeed $(M(n)f)(s)=U(n)f(\rho (n{)}^{-1}s)=U(n)U(n{)}^{-1}f(s)=f(s)$ because $f$ is $N$-linear; and for $i\in I$, $n\in N$ the identities $P(in)=P(i)\rho (n)$ and $P(ni)=\rho (n)P(i)$ of [F1] give $(M(in)f)(s)=U(i)U(n)f(\rho (n{)}^{-1}P(i{)}^{-1}s)=U(i)f(P(i{)}^{-1}s)=(M(i)f)(s)$, while $(M(ni)f)(s)=U(n)U(i)f(P(i{)}^{-1}\rho (n{)}^{-1}s)=U(n)U(i)U(i^{-1}{n}^{-1}i)f(P(i{)}^{-1}s)=U(i)f(P(i{)}^{-1}s)=(M(i)f)(s)$. Hence $M$ descends to a well-defined map $M:Q\to {\mathrm{End}}_{\mathbb{C}}(M)$, written $M(q):=M(i)$ for any $i\in I$ with $iN=q$.

For all $i,j\in I$ one has $M(i)M(j)=\alpha (iN,jN{)}^{-1}M(ij)$: indeed $(M(i)M(j)f)(s)=U(i)(M(j)f)(P(i{)}^{-1}s)=U(i)U(j)f(P(j{)}^{-1}P(i{)}^{-1}s)=U(i)U(j)f((P(i)P(j){)}^{-1}s)$, and $P(i)P(j)=\alpha (iN,jN)P(ij)$ by [F1], so $(P(i)P(j){)}^{-1}=\alpha (iN,jN{)}^{-1}P(ij{)}^{-1}$ and the last expression equals $\alpha (iN,jN{)}^{-1}U(ij)f(P(ij{)}^{-1}s)=\alpha (iN,jN{)}^{-1}(M(ij)f)(s)$.

The evaluation $E_U:S{\otimes }_{\mathbb{C}}M\to U$, $E_U(s\otimes f)=f(s)$, is an $I$-isomorphism: it is an $N$-isomorphism by [F6] applied to the $\theta$-isotypical module $U$ of step 1.1, and it is $I$-equivariant because $E_U(P(i)s\otimes M(i)f)=(M(i)f)(P(i)s)=U(i)f(P(i{)}^{-1}P(i)s)=U(i)E_U(s\otimes f)$ for all $i\in I$.

By step 3.1 and the identities of [F1], $M(1)=M(N)={\mathrm{id}}_M$, and every $M(i)$ is invertible: from step 3.2 with $j=i^{-1}$ one gets $M(i)M(i^{-1})=\alpha (iN,i^{-1}N{)}^{-1}M(1)=\alpha (iN,i^{-1}N{)}^{-1}{\mathrm{id}}_M$, and symmetrically $M(i^{-1})M(i)=\alpha (i^{-1}N,iN{)}^{-1}{\mathrm{id}}_M$, so $M(i{)}^{-1}=\alpha (iN,i^{-1}N)M(i^{-1})$.

A subspace $M_0\subseteq M$ is $M$-stable, that is $M(q)M_0\subseteq M_0$ for all $q\in Q$, if and only if $E_U(S\otimes M_0)$ is an $I$-submodule of $U$. If $M(q)M_0\subseteq M_0$ for all $q$, then for $i\in I$ one has $i\cdot E_U(S\otimes M_0)=E_U(P(i)S\otimes M(i)M_0)=E_U(S\otimes M(i)M_0)\subseteq E_U(S\otimes M_0)$ by step 3.3. Conversely, if $U_0\subseteq U$ is an $I$-submodule, then $U_0$ is an $N$-submodule of the $\theta$-isotypical module $U$, so $U_0=E_U(S\otimes M_0)$ for the unique $M_0={\mathrm{Hom}}_N(S,U_0)\subseteq M$ by [F6]; for $f\in M_0$ the element $M(i)f$ has image $U(i)f(S)\subseteq U(i)U_0\subseteq U_0$, so $M(i)M_0\subseteq M_0$. Since $M_0\mapsto S\otimes M_0$ is injective and $\dim E_U(S\otimes M_0)=\dim S\cdot \dim M_0$, these two assignments are mutually inverse bijections between the $M$-stable subspaces of $M$ and the $I$-submodules of $U$; hence $U$ is irreducible if and only if $M$ is, and then $M$ is an irreducible projective $Q$-representation with factor set ${\alpha }^{-1}$ by steps 3.2 and 4.1 and definition [F3], equivalently an irreducible left ${\mathbb{C}}^{{\alpha }^{-1}}[Q]$-module with the same invariant subspaces by [F4].

Conversely let $M:Q\to \mathrm{GL}(M_0)$ be an irreducible projective $Q$-representation with factor set ${\alpha }^{-1}$ on a nonzero finite-dimensional space $M_0$, and put $U:=S{\otimes }_{\mathbb{C}}{M}_0$ with $i\cdot (s\otimes m):=P(i)s\otimes M(iN)m$. This is a well-defined $I$-action: it is bilinear, $1$ acts as $\mathrm{id}$ by [F1], and $i\cdot (j\cdot (s\otimes m))=P(i)P(j)s\otimes M(iN)M(jN)m=\alpha (iN,jN)P(ij)s\otimes \alpha (iN,jN{)}^{-1}M(ijN)m=(ij)\cdot (s\otimes m)$ by the defining relations of $P$ and $M$. Since $M(n)=\mathrm{id}$, the restriction to $N$ is $n\cdot (s\otimes m)=\rho (n)s\otimes m$, so $U{\mathrm{\mid }}_N\cong (\dim M_0)\cdot \rho$ and $U$ is a $\theta$-isotypical $N$-module lying over $\theta$. Finally $U$ is irreducible: by [F6] every $N$-submodule $U_0\subseteq U$ is $E_U(S\otimes M_1)$ for a unique $M_1={\mathrm{Hom}}_N(S,U_0)\subseteq M_0$, and if $U_0$ is $I$-stable then $M(iN)M_1\subseteq M_1$ for all $i$ by the computation of step 5.1, so $M_1=0$ or $M_1=M_0$ by irreducibility of $M$, whence $U_0=0$ or $U_0=U$.

The two constructions are inverse on isomorphism classes. Starting from $U$, forming $M={\mathrm{Hom}}_N(S,U)$ and then $S\otimes M$ with the action of step 6.1, the evaluation $E_U$ is an $I$-isomorphism by step 3.3, so $S\otimes M\cong U$. Starting from $M_0$, forming $U=S\otimes M_0$ and then ${\mathrm{Hom}}_N(S,U)$, the $N$-maps $s\mapsto s\otimes m$ identify ${\mathrm{Hom}}_N(S,U)$ with $M_0$ by [F6], and the induced operator sends $f_m$ to $f_{M(iN)m}$ because $(M^{\mathrm{\prime }}(iN)f_m)(s)=U(i)f_m(P(i{)}^{-1}s)=P(i)P(i{)}^{-1}s\otimes M(iN)m=s\otimes M(iN)m$; so the isomorphism classes correspond. Both assignments send isomorphisms to isomorphisms, since an $I$-isomorphism $U\to U^{\mathrm{\prime }}$ restricts to an isomorphism ${\mathrm{Hom}}_N(S,U)\to {\mathrm{Hom}}_N(S,U^{\mathrm{\prime }})$ and a $Q$-isomorphism $M_0\to M_0^{\mathrm{\prime }}$ induces $S\otimes M_0\to S\otimes M_0^{\mathrm{\prime }}$, and the constructions of steps 5.1 and 6.1 preserve irreducibility in both directions; hence $U\mapsto M$ is a bijection from the isomorphism classes of irreducible $I$-modules lying over $\theta$ onto those of irreducible projective $Q$-representations with factor set ${\alpha }^{-1}$.

Composing the bijection of step 7.1 with induction to $G$ gives the required bijection onto $\mathrm{Irr}(G\mid \theta )$: by [F7] induction is a bijection $\mathrm{Irr}(I\mid \theta )\to \mathrm{Irr}(G\mid \theta )$ whose inverse takes the $\theta$-isotypical component, and [F8] identifies that inverse explicitly through the irreducible $I$-module $W=V_{\theta }$ and the canonical isomorphism ${\mathrm{Ind}}_I^{G}W\to V$. Moreover, for a fixed $G$-orbit of $\mathrm{Irr}(N)$ the target set $\mathrm{Irr}(G\mid \theta )$ is the same for every representative $\theta$ of the orbit, and the sets belonging to distinct orbits partition $\mathrm{Irr}(G)$ by [F7]; so choosing one $\theta$ per orbit lists every irreducible $G$-representation exactly once.

Two degenerate cases match the statement. If $I=N$, then $Q$ is the trivial group, the only normalized two-cocycle on it is the constant function $1$, the only irreducible projective $Q$-representation is the trivial one-dimensional module, and the construction of step 6.1 returns $M_0=\mathbb{C}$ with $U=S\otimes \mathbb{C}\cong S$ and $i\cdot (s\otimes m)=\rho (i)s\otimes m$ for $i\in N$, so $\mathrm{Irr}(I\mid \rho )=\{\theta \}$: the quotient contributes its unique trivial module. If instead $\alpha$ is trivializable, then $\rho$ extends to $I$ by [F10]; choosing the operators $P$ to be such an extension $\tilde{S}$ gives $\alpha =1$, so step 3.2 makes $M$ an ordinary representation of $Q=I\mathrm{/}N$ and the $I$-action of step 6.1 is $i\cdot (s\otimes m)=\tilde{S}(i)s\otimes M(iN)m$, whose character is ${\chi }_{\tilde{S}}{\mathrm{Inf}}_{I\mathrm{/}N}^I{\chi }_M$, exactly the parametrization of [F9]; thus the theorem reduces to Gallagher's correspondence in the extendible case, and only the nonvanishing of the obstruction makes the projective version necessary.

Steps 7.1 and 8.1 establish the bijection from the irreducible projective $Q$-representations with factor set ${\alpha }^{-1}$ to $\mathrm{Irr}(I\mid \rho )$ given by $M\mapsto S\otimes M$ and $U\mapsto {\mathrm{Hom}}_N(S,U)$, and its composition with induction onto $\mathrm{Irr}(G\mid \theta )$, with the orbit bookkeeping for $\theta$; step 9.1 disposes of the cases $I=N$ and $\alpha$ trivializable. This is precisely the correspondence asserted, valid for the nonsplit case in which the Clifford obstruction is nonzero.