Principal Series Representations of GL N over a Finite Field — Examples

By [F1] applied to $S_2=\{1,s\}$ the algebra $H$ has the $\mathbb{C}$-basis $T_1=e_B$ and $T_s=q{e}_B\dot{s}{e}_B=\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{x\in B\dot{s}B}x$, so ${\dim }_{\mathbb{C}}H=2$.

The algebra $H$ is generated by $T_s$, because the unital algebra generated by $T_s$ contains the unit $T_1$ and every element is a combination of the two basis elements; it is commutative because $T_s^2=(q-1)T_s+q{T}_1$ lies in the span of $1$ and $T_s$ by [F2]. Hence evaluation $\mathbb{C}[T]\to H$, $T\mapsto T_s$, is a surjective unital algebra homomorphism whose kernel contains $T^2-(q-1)T-q$; the induced map $\mathbb{C}[T]\mathrm{/}(T^2-(q-1)T-q)\to H$ is a surjection from a $2$-dimensional algebra onto a $2$-dimensional algebra, hence an isomorphism. In particular the displayed multiplication rule of [F2] is the complete multiplication table of $H$ in this basis, matching the presentation of [F3].

Factoring $T^2-(q-1)T-q=(T-q)(T+1)$ and noting that $q\mathrm{\ne }-1$ because $q\ge 2$, the ideals $(T-q)$ and $(T+1)$ of $\mathbb{C}[T]$ are comaximal, so [F5] gives an isomorphism $\mathbb{C}[T]\mathrm{/}((T-q)(T+1))\cong \mathbb{C}[T]\mathrm{/}(T-q)\times \mathbb{C}[T]\mathrm{/}(T+1)\cong \mathbb{C}\oplus \mathbb{C}$, hence $H\cong \mathbb{C}\oplus \mathbb{C}$ by step 2.1. Under this isomorphism the two factors are the two simple $H$-modules and the two orthogonal idempotents of $H$ are the corresponding projections; for $q=2$ the factorization reads $(T-2)(T+1)$.

By [F4] right multiplication identifies $H$ with ${\mathrm{End}}_G(\mathbb{C}[G\mathrm{/}B]{)}^{\mathrm{o}\mathrm{p}}$ and $\mathbb{C}[G\mathrm{/}B]\cong \mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)]$, so ${\dim }_{\mathbb{C}}{\mathrm{End}}_G(\mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)])={\dim }_{\mathbb{C}}H=2$ by step 1.1; since $H$ is commutative by step 2.1, the opposite algebra is $H$ itself and the opposite-algebra ambiguity is invisible.

Step 1.1 gives the basis $T_1=e_B$, $T_s$ and the dimension, step 2.1 gives the quadratic multiplication and the presentation $\mathbb{C}[T]\mathrm{/}(T^2-(q-1)T-q)$, step 3.1 gives the splitting $H\cong \mathbb{C}\oplus \mathbb{C}$, and step 3.2 gives the endomorphism-algebra dimension and the opposite-algebra statement. All algebras are finite-dimensional over $\mathbb{C}$, all sums are finite, and no choice principle is used.

By [F1] the module $\mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)]$ is $I(1)$ of dimension $q+1$. By [F2] it splits as $\mathbf{1}\oplus \mathrm{St}$ with $\dim \mathrm{St}=q$, both constituents of multiplicity one, and the standard intertwiner acts by $q$ on $\mathbf{1}$ and by $-1$ on $\mathrm{St}$.

By [F5] the Hecke algebra $H\cong \mathbb{C}\oplus \mathbb{C}$ has exactly two simple modules, and $T_s$ acts on them by the scalars $q$ and $-1$; these match the two constituents of step 1.1 through the identification ${\mathrm{End}}_G(\mathbb{C}[G\mathrm{/}B])\cong H^{\mathrm{o}\mathrm{p}}\cong H$.

By [F3] the constituents of the spherical principal series are indexed by the partitions $\lambda ⊢2$, namely $(2)$ and $(1,1)$, and by [F4] both occur with multiplicity $f^{(2)}=f^{(1,1)}=1$, agreeing with the multiplicity-one splitting of step 1.1. Under the noncanonical Tits parametrisation we may choose the matching so that the Hecke character $T_s\mapsto q$ labels the trivial constituent $\mathbf{1}$ and $T_s\mapsto -1$ labels the sign character, hence $\mathbf{1}$ corresponds to $(2)$ and $\mathrm{St}$ to $(1,1)$.

Steps 1.1, 2.1 and 3.1 give the splitting $\mathbb{C}[{\mathbb{P}}^1]=\mathbf{1}\oplus \mathrm{St}$ with $\dim \mathrm{St}=q$, the multiplicity-one statement, the $T_s$-eigenvalues $q$ and $-1$, and the partition labels under the noncanonical parametrisation. AC is carried only from the Tits-deformation supplier [F6], as declared.

For $q=2$ the group ${\mathbb{F}}_2^{\times }=\{1\}$ is trivial, so the diagonal torus $T\cong ({\mathbb{F}}_2^{\times }{)}^n$ is the trivial group and its character group has the single element $1$; the Weyl action fixes it, so $W_1=S_n$, and every principal series is $I(1)$. A regular character would need pairwise distinct coordinates, impossible in a one-element group when $n\ge 2$; for $n=1$ the single coordinate is vacuously pairwise distinct.

Since the unique character is fixed by $W$, [F3] gives ${\dim }_{\mathbb{C}}{\mathrm{End}}_G(I(1))=n!$, and [F4] gives the constituent indexing by partitions with multiplicities $f^{\lambda }$.

By [F5] the Hecke algebra is semisimple with quadratic relation $T_{s_i}^2=T_{s_i}+2$ at $q=2$, and the two roots $2\mathrm{\ne }-1$ are distinct; by [F6] Tits deformation identifies it with $\mathbb{C}[S_n]$ in this case as in every other. Hence the collapse of $\widehat{T}$ to a point and of the Weyl action to the trivial action is a genuine boundary phenomenon of the parametrising torus, while the endomorphism algebra, the constituent multiplicities and the Tits isomorphism retain their general form.

Steps 1.1-1.3 establish the two boundary statements: the torus and the regular characters collapse, whereas the endomorphism algebra, the partition parametrisation with multiplicities $f^{\lambda }$ and the Tits isomorphism to $\mathbb{C}[S_n]$ remain valid. AC is carried only from the Tits-deformation supplier [F6], as declared.

By [F1] every principal series of ${\mathrm{GL}}_3({\mathbb{F}}_q)$ has dimension $[G:B]=(q+1)(q^2+q+1)$.

Case (a): if the three coordinates are pairwise distinct, no nontrivial permutation fixes the character, so $W_{\chi }=1$; by [F2] $\dim {\mathrm{End}}_G(I(\chi ))=1$ and $I(\chi )$ is irreducible.

Case (b): if $\chi =(a,a,b)$ with $a\mathrm{\ne }b$, the stabiliser is $W_{\chi }\cong S_2$, so $\dim {\mathrm{End}}_G(I(\chi ))=2$ by [F2]. The equal-coordinate blocks are $n_1=2,n_2=1$, so by [F3] the constituents are indexed by the tuples $({\lambda }^{(1)},{\lambda }^{(2)})$ with ${\lambda }^{(1)}⊢2$, ${\lambda }^{(2)}⊢1$, and have multiplicities $f^{{\lambda }^{(1)}}{f}^{{\lambda }^{(2)}}$; by [F4] these multiplicities are all $1$, so there are exactly two non-isomorphic constituents, each of multiplicity one, and ${\mathrm{End}}_G(I(\chi ))\cong \mathbb{C}\oplus \mathbb{C}$.

Put $\psi =a\circ \det$ and let $W=\mathbb{C}{z}_0$ with $g{z}_0=\psi (g)z_0$. Multiplicativity and the triangular determinant formula show that $\psi$ is a character and $\psi {\mathrm{\mid }}_B$ is the inflation of $(a,a,a)$ (For same-sized finite square matrices over a commutative ring, $\det (AB)=\det (A)\det (B)$, The determinant of a triangular matrix is the product of its diagonal entries). Define $\mathrm{\Phi }:I(1)\otimes W\to I(a,a,a)$ by $\mathrm{\Phi }(f\otimes z_0)(x)=f(x)\psi (x{)}^{-1}$. This lands in $I(a,a,a)$ because $f(xb)=f(x)$ and $\psi (xb)=\psi (x)\psi (b)$, and it is $G$-equivariant since $\mathrm{\Phi }(g\cdot (f\otimes z_0))(x)=f(g^{-1}x)\psi (g)\psi (x{)}^{-1}=\mathrm{\Phi }(f\otimes z_0)(g^{-1}x)$ by [F6]. Its inverse sends $F$ to $(x\mapsto F(x)\psi (x))\otimes z_0$, whose function is right $B$-invariant. Thus $I(a,a,a)\cong I(1)\otimes (a\circ \det )$.

Case (c): if $\chi =(a,a,a)$, then $W_{\chi }=S_3$ and $\dim {\mathrm{End}}_G(I(\chi ))=6$ by [F2]; by [F3] the constituents are indexed by the partitions $\lambda ⊢3$ with multiplicities $f^{\lambda }$, which by [F4] are $1,2,1$ for $(3),(2,1),(1,1,1)$, and ${\mathrm{End}}_G(I(\chi ))\cong \mathbb{C}\times {\mathrm{M}}_2(\mathbb{C})\times \mathbb{C}\cong \mathbb{C}[S_3]$. Moreover the twist isomorphism of step 1.4 with $\psi =a\circ \det$ gives $I(a,a,a)\cong I(1)\otimes (a\circ \det )$; tensoring with a one-dimensional character preserves dimensions and multiplicities, so the constituent count and multiplicities agree with the spherical computation [F5] after twisting.

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 give the dimension, the three cases (a)-(c) with their endomorphism dimensions and constituent multiplicities, and the agreement with the general corollary and with the twisted spherical computation in case (c). AC is carried only from the Tits-deformation supplier in [F7], as declared; all modules are finite-dimensional over $\mathbb{C}$.