1 Introduction
The restriction problem, which examines how representations behave when restricted from a group to a subgroup, is a fundamental topic in representation theory. In general, the restriction of an irreducible representation of a group to a subgroup is neither irreducible nor completely reducible. The general theory by which the restriction of an irreducible representation of a group decomposes into irreducible representations of its subgroup is explained by branching rules. One is interested in computing the multiplicity with which an irreducible representation of the subgroup appears (as a subquotient) in the restriction of an irreducible representation of the given group. More generally, one may ask how a member of a class of representations of a group decomposes under restriction into member of this class for the subgroup. In Kobayashi [Reference Kobayashi9], the author gives a survey of branching problems for complex representations of real reductive groups. See also, for instance, [Reference Cheng4, Reference Gan, Gross and Prasad5, Reference Paškūnas10, Reference Reeder12] for other aspects of the restriction problem.
The decomposition of a finite tensor product of representations of a group G into its irreducible representations can be considered as part of a restriction problem as it can be seen as the restriction of the representation of the group
$G\times G\times \cdots \times G$
(finitely many times) to the diagonal subgroup which is isomorphic to G. In this case, the branching is explained by what are called Clebsch–Gordan formulas. This situation occurs naturally in the study of the restriction of irreducible mod p representations of finite groups of Lie type, and in particular in the study of the restriction of irreducible mod p representations (weights) of
$\text {GL}_2({\mathbb {F}_q})$
to
$\text {GL}_2({\mathbb {F}_p})$
, where
${\mathbb {F}_q} = {\mathbb F}_{p^f}$
is a finite extension of degree
$f \geq 1$
of the finite field
${\mathbb {F}_p}$
with p elements. See, for instance, [Reference Gupta8] for an explicit decomposition in the case that
${\mathbb {F}_q}$
is a degree
$f = 3$
extension of
${\mathbb {F}_p}$
using the Clebsch–Gordan formula from [Reference Glover7] for
$\text {GL}_2({\mathbb {F}_p})$
. See also [Reference Bhattacharjee, Ghate, Pandey and Veerapaneni2] for Clebsch–Gordan formulas for
$\text {GL}_2({\mathbb {F}_q})$
(which would be useful for solving the restriction problem when the group that one is restricting to is
$\text {GL}_2({\mathbb {F}_q})$
, cf. Remark 3.11).
In this context, it would be interesting to study the restriction problem for a slightly more complicated but natural class of representations of these groups. Let 
and 
. Let
${B_p}$
and
${B_q}$
denote the Borel subgroups of all upper triangular matrices of
${G_p}$
and
${G_q}$
, respectively. A natural problem would be to study the restriction to
${G_p}$
of principal series representations of
${G_q}$
obtained by inducing a mod p character of
${B_q}$
. Similarly, let
$T_p$
and
$T_q$
denote the anisotropic torus of
${G_p}$
and
${G_q}$
, respectively. One could also study the restrictions from
${G_q}$
to
${G_p}$
of representations induced from a mod p character of
${T_q}$
. The latter representations are closely connected to the mod p reductions of complex cuspidal representations of
${G_q}$
.
In this article, we solve the restriction problem for both these classes of representations of
$G_q$
completely, by decomposing their restrictions to
$G_p$
into a direct sum of analogous representations of
${G_p}$
(possibly twisted by the mod p Steinberg representation of
${G_p}$
). The explicit decompositions are contained in Theorems 2.18 and 3.9 which are the main results of this article. The key tool is Mackey’s restriction formula. However, to use it, one needs to make the orbit decomposition of
${G_p} \backslash {G_q} / H$
reasonably explicit where
$H = {B_q}$
or
${T_q}$
. This is done using the projective line as a crutch, noting that
${G_q}/{B_q}\simeq {\mathbb {P}^1({\mathbb {F}_q})}$
and
${G_q}/{T_q}$
is the orbit of an element in
${\mathbb {P}^1({\mathbb {F}_{q^2}})}$
outside
${\mathbb {P}^1({\mathbb {F}_q})}$
. These orbit decompositions may be of independent interest.
2 Restriction of principal series representations
Recall that for a mod p character
$\chi $
of
${B_q}$
, the mod p principal series representation of
${G_q}$
is the induced representation
$\text {ind}_{{B_q}}^{{G_q}} \chi $
. Its Jordan–Hölder factors are described in [Reference Bardoe and Sin1, Reference Breuil and Paškūnas3]. While a complete decomposition of the restriction of such a principal series representation of
${G_q}$
to
${G_p}$
is not immediately evident, one might expect that the principal series
$\mathrm {ind}_{{B_p}}^{{G_p}}({\chi |}_{{B_p}})$
of
$G_p$
(which has two Jordan–Hölder factors) be contained in the restriction to
${G_p}$
.
Lemma 2.1 We have
$\mathrm {ind}_{{B_p}}^{{G_p}}\left ({\chi |}_{{B_p}}\right ) \hookrightarrow {\left .\mathrm {ind}_{{B_q}}^{{G_q}}\chi \right |}_{{G_p}}.$
Proof By Frobenius reciprocity, this is equivalent to checking whether
${\chi |}_{{B_p}}\hookrightarrow {\left .\text {ind}_{{B_q}}^{{G_q}}{\chi }\right |}_{{B_p}}$
, which is true as
$\chi \hookrightarrow {\left .\text {ind}_{{B_q}}^{{G_q}}{\chi }\right |}_{{B_q}}$
(again, by Frobenius reciprocity).
The real question then is what the other representations appearing in the decomposition of the restriction are. In this section, we shall give a complete answer in terms of principal series representations, the Steinberg representation, and cuspidal representations of
${G_p}$
.
2.1 The action of
${G_p}$
on
${\mathbb {P}^1({\mathbb {F}_q})}$
There is a natural action of
${G_q}$
on the projective space
${\mathbb {P}^1({\mathbb {F}_q})}$
given by
for
$\left (\begin {matrix} a & b \\ c & d \end {matrix}\right ){\kern-1pt}\in{\kern-1pt} {G_q}$
and
$[x{\kern-1pt}:{\kern-1pt}y]{\kern-1pt}\in{\kern-1pt} {\mathbb {P}^1({\mathbb {F}_q})}$
. This action is transitive with
${{\text {Stab}_{{G_q}}}([1{\kern-1pt}:{\kern-1pt}0]){\kern-1pt}={\kern-1pt}{B_q}}$
. We identify
${\mathbb {P}^1({\mathbb {F}_q})}$
with
$\{[1:x]:x\in {\mathbb {F}_q}\}\cup \{[0:1]\}$
throughout the article.
Notation 2.2 Denote ![x-hat is defined as [1 : x]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20260918022736332-0384:S0008439526101623:S0008439526101623_inline73.png?pub-status=live){kind=small}
and ![Infinity hat is defined as [0 : 1]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20260918022736332-0384:S0008439526101623:S0008439526101623_inline74.png?pub-status=live){kind=small}
.
Notation 2.3 Fix
$\epsilon \in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_q}\text { such that }\epsilon ^2\in {\mathbb {F}_q}$
. Denote the anisotropic torus of
${G_q}$
by
Analogously, for fixed
$\eta \in {\mathbb {F}_{p^2}}\setminus {\mathbb {F}_p}\text { such that }\eta ^2\in {\mathbb {F}_p}$
, let
denote the anisotropic torus of
${G_p}$
.
Lemma 2.4 Under the action of
${G_q}$
, we have the orbit decomposition
Moreover, the stabilizers of
$\kern2pt{\widehat {0}}$
and
${\widehat {\epsilon }}$
are equal to
${B_q}$
and
$T_q$
, respectively.
Proof The transitivity of the
${G_q}$
action on
${\mathbb {P}^1({\mathbb {F}_q})}$
gives the orbit
${O_{{G_q}}}({\widehat {0}})={\mathbb {P}^1({\mathbb {F}_q})}=\{{\widehat {x}}:x\in {\mathbb {F}_q}\}\sqcup \{\widehat {\infty }\}$
. That
${\text {Stab}_{{G_q}}}({\widehat {0}})={B_q}$
can easily be verified. Note that
${\widehat {\epsilon }}\notin {O_{{G_q}}}({\widehat {0}})$
since
$\epsilon \in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_q}$
. If we show that
$\left |{\mathbb {P}^1({\mathbb {F}_{q^2}})}\right |=\left |{O_{{G_q}}}({\widehat {0}})\sqcup {O_{{G_q}}}({\widehat {\epsilon }})\right |$
, then we are done. Let
$g=\left (\begin {matrix} a & b \\ c & d \end {matrix}\right )\in {\text {Stab}_{{G_q}}}({\widehat {\epsilon }})$
, that is,
$g\cdot {\widehat {\epsilon }}={\widehat {\epsilon }}$
. Then, for some
$0\neq \lambda \in {\mathbb {F}_{q^2}}, ~(a+b\epsilon ,c+d\epsilon )=(\lambda ,\lambda \epsilon ) \text { which gives }(a+b\epsilon )\epsilon =c+d\epsilon $
. As
$1,\epsilon $
are linearly independent over
${\mathbb {F}_q}$
, we get
$a=d$
and
$c=b\epsilon ^2$
. Thus,
${\text {Stab}_{{G_q}}}({\widehat {\epsilon }})=T_q$
and so the number of elements in
${O_{{G_q}}}({\widehat {\epsilon }})$
is
$\frac {\left (q^2-1\right )\left (q^2-q\right )}{q^2-1}=q^2-q$
and
$\left |{O_{{G_q}}}({\widehat {0}})\right |+\left |{O_{{G_q}}}({\widehat {\epsilon }})\right |=q^2+1=\left |{\mathbb {P}^1({\mathbb {F}_{q^2}})}\right |$
.
Notation 2.5 For
$f>2$
, define the set
$X=\left \{\begin {array}{@{}ll} {\mathbb {F}_q}\setminus {\mathbb {F}_p}, & \text {if } 2\nmid f, \\ {\mathbb {F}_q}\setminus {\mathbb {F}_{p^2}}, &\text {if }2\mid f. \end {array}\right .$
Lemma 2.6 For all
$x\in X$
, the set
$\{1,x,x^2\}$
is linearly independent over
${\mathbb {F}_p}$
.
Proof Let
$x\in {\mathbb {F}_q}$
and let
$\{1,x,x^2\}$
be linearly dependent over
${\mathbb {F}_p}$
. Then, there exist
$a_0,a_1,a_2\in {\mathbb {F}_p}$
, not all zero, such that
$a_0+a_1x+a_2x^2=0$
. That is, x satisfies a polynomial of degree at most
$2$
over
${\mathbb {F}_p}$
. If either
$a_2=0$
or the polynomial is reducible, then
$x\in {\mathbb {F}_p}$
. If
$a_2\neq 0$
and the polynomial is irreducible, then
${\mathbb {F}_{p^2}}\simeq {\mathbb {F}_p}(x)\hookrightarrow {\mathbb {F}_q}$
. But
${\mathbb {F}_{p^2}}\hookrightarrow {\mathbb {F}_q}$
iff
$2\mid f$
. Thus,
$x\in {\mathbb {F}_p}$
when
$2\nmid f$
and
$x\in {\mathbb {F}_{p^2}}$
when
$2\mid f$
.
Lemma 2.7 With respect to the
${G_p}$
action on
${\mathbb {P}^1({\mathbb {F}_q})}$
,
where
$Z_p$
is the center of
${G_p}$
.
Proof Let
$\left (\begin {matrix} a & b \\ c & d \end {matrix}\right ){\kern-1pt}\in{\kern-1pt} {\text {Stab}_{{G_p}}}({\widehat {x}})$
, then
$(a{\kern-1pt}+{\kern-1pt}bx,c{\kern-1pt}+{\kern-1pt}dx)=(\lambda ,\lambda x)$
, for some
${0{\kern-1pt}\neq{\kern-1pt} \lambda {\kern-1pt}\in{\kern-1pt} {\mathbb {F}_q}}$
. So, we get
$c+(d-a)x-bx^2=0$
and by Lemma 2.6,
$a=d$
and
$b=c=0$
.
Proposition 2.8 The decomposition of
${\mathbb {P}^1({\mathbb {F}_q})}$
into its
${G_p}\text {-orbits}$
is given as
where
$\eta \in {\mathbb {F}_{p^2}}\setminus {\mathbb {F}_p}$
with
$\eta ^2\in {\mathbb {F}_p}$
and where
${\mathcal {I}_1}$
and
${\mathcal {I}_2}$
form a complete set of representatives of the
${G_p}\text {-orbits}$
in
${\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_p})}$
and
${\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_{p^2}})}$
, respectively. Also,
Proof Suppose
$2\nmid f$
. Since
${\mathbb {P}^1({\mathbb {F}_p})}\hookrightarrow {\mathbb {P}^1({\mathbb {F}_q})}$
is
${G_p}\text {-stable,}$
we have
Note that
${G_p}\backslash {\mathbb {P}^1({\mathbb {F}_p})}=\{{O_{{G_p}}}({\widehat {0}})\}$
as
${G_p}$
acts transitively on
${\mathbb {P}^1({\mathbb {F}_p})}$
. Moreover, for any
${\widehat {x}}\in {\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_p})},~ {\text {Stab}_{{G_p}}}({\widehat {x}})=Z_p$
by Lemma 2.7, and so
$\left |{O_{{G_p}}}({\widehat {x}})\right |=p\left (p^2-1\right )$
. Since the orbit of each element in
${\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_p})}$
has the same cardinality, the number of
${G_p}\text {-orbits}$
in
${\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_p})}$
is
Suppose
$2\mid f$
. Since
${\mathbb {F}_{p^2}}\hookrightarrow {\mathbb {F}_q}$
and
${\mathbb {P}^1({\mathbb {F}_{p^2}})}\hookrightarrow {\mathbb {P}^1({\mathbb {F}_q})}$
is
${G_p}\text {-stable}$
, we have
Note that
${G_p}\backslash {\mathbb {P}^1({\mathbb {F}_{p^2}})} = \{{O_{{G_p}}}({\widehat {0}}),{O_{{G_p}}}({\widehat {\eta }})\}$
by taking
$f=1$
in Lemma 2.4. Again, for any
${\widehat {x}}\in {\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_{p^2}})},~ {\text {Stab}_{{G_p}}}({\widehat {x}})\simeq Z_p$
by Lemma 2.7, and so,
$\left |{O_{{G_p}}}({\widehat {x}})\right |=p\left (p^2-1\right )$
. So, by the same argument as in the previous case, the number of
${G_p}\text {-orbits}$
in
${\mathbb {P}^1({\mathbb {F}_q})}\setminus {\mathbb {P}^1({\mathbb {F}_{p^2}})}$
is
2.2 Decomposition of the restriction
For a mod p character
$\chi $
of
${B_q}$
, let
$\text {ind}_{{B_q}}^{{G_q}}\chi $
be a principal series representation of
${G_q}$
. Then, by Mackey’s restriction formula, we have
where 
and
${\chi ^\gamma }(b)=\chi \left (\gamma ^{-1}b\gamma \right )$
for all
$b\in {B_q^\gamma }$
and where
$\Gamma $
is a set of double coset representatives of
${G_p}\backslash {G_q}/{B_q}$
.
Notation 2.9 For
$x\in {\mathbb {F}_q}$
, let 
and 
.
Remark 2.10 A complete set of coset representatives of
${G_q}/{B_q}$
is given by
${\{g_x:x\in {\mathbb {F}_q}\}\cup \{w\}}$
, and under the identification
${G_q}/{B_q}\simeq {\mathbb {P}^1({\mathbb {F}_q})}$
, we see that
$g_x{B_q}\mapsto {\widehat {x}}\text { and }w{B_q}\mapsto \widehat {\infty }$
.
Under the action of
${G_p}$
on
${G_q}/{B_q}$
, we have
${\text {Stab}_{{G_p}}}\left (g{B_q}\right )=g {B_q} g^{-1}\cap {G_p} = B_q^g$
. Hence, taking
$g=g_x$
and using Remark 2.10, we have
Now, the double cosets in
${G_p}\backslash {G_q}/{B_q}$
are in one-to-one correspondence with the orbits in
${G_p}\backslash \left ({G_q}/{B_q}\right )$
and hence with the orbits in
${G_p}\backslash {\mathbb {P}^1({\mathbb {F}_q})}$
by Remark 2.10. Also,
Moreover,
${G_p} w{B_q}={G_p} g_0{B_q}$
. Thus, if
$\Gamma $
is a set of double coset representatives of
${G_p}\backslash {G_q}/{B_q}$
, then we may assume
$\Gamma \subseteq \{g_x: x\in {\mathbb {F}_q}\}$
. Again, by Remark 2.10,
Thus, substituting (2.4) and (2.5) in (2.3), we get
Now, Proposition 2.8 gives the description of all the
${G_p}\text {-orbits}$
in
${\mathbb {P}^1({\mathbb {F}_q})}$
and therefore,
Note that, by Lemma 2.4, the stabilizers of
${\widehat {0}}$
and
${\widehat {\eta }}$
in
${G_p}$
are
${B_p}$
and
$T_p$
, respectively, and
$\chi ^{g_0}={\chi |}_{{B_p}}$
. Also, for all
${\widehat {x}}$
in
${\mathcal {I}_1}$
or
${\mathcal {I}_2}$
, the corresponding field element x is in X (see the definition of
${\mathcal {I}_1}$
and
${\mathcal {I}_2}$
) and so,
${\text {Stab}_{{G_p}}}({\widehat {x}})=Z_p$
, for all
${\widehat {x}}$
in
${\mathcal {I}_1}$
or
${\mathcal {I}_2}$
by Lemma 2.7. Lastly, we note that
${\left .\chi ^{g_x}\right |}_{Z_p} ={\chi |}_{Z_p}$
, for all
$g_x\in {G_q}$
and so
$\text {ind}_{{\text {Stab}_{{G_p}}}({\widehat {x}})}^{G_p} \chi ^{g_x}=\text {ind}_{Z_p}^{G_p} \left ({\chi |}_{Z_p}\right )$
, for all
${\widehat {x}}\in {\mathcal {I}_1}$
or
${\widehat {x}}\in {\mathcal {I}_2}$
. Combining this information, we get
where
$nV$
denotes the direct sum of n copies of the representation V.
Notation 2.11 For all positive integers f, any character of
${B_q}$
is of the form
$\chi _{r,s}$
for
$0\leq r,s<q-1$
, where
Denote 
.
Since
$\text {ind}_{{B_q}}^{{G_q}}\left (\chi _{r,s}\right )\simeq \text {ind}_{{B_q}}^{{G_q}}\left (\chi _{r-s}\otimes \text {det}^s\right )\simeq \text {ind}_{{B_q}}^{{G_q}}\chi _{r-s}\otimes \text {det}^s$
, for all
$r,s\in \{0,1,\ldots , q-2\}$
, it is sufficient to analyze the restriction
${\left .\text {ind}_{{B_q}}^{{G_q}}\chi _r\right |}_{{G_p}}$
.
Notation 2.12 The split torus of
${G_p}$
is denoted by 
.
Notation 2.13 Let
$f \geq 1$
. The character
$\omega _{2f} : T_q \rightarrow {\mathbb F}_{q^2}^*$
is defined by
All the characters of
$T_q$
are then precisely of the form
$\omega _{2f}^r$
, where
$0\leq r<q^2-1$
.
In particular, the character
$\omega _2$
of
$T_p$
is given as
Lemma 2.14 If f is even, then as characters of
$T_p$
, we have
$\chi _r^{g_{\eta }}=\omega _2^r$
.
Proof Indeed, for all
$\left (\begin {matrix} a & b \\ b\eta ^2 & a \end {matrix}\right )\in T_p$
, we have
Notation 2.15 For a complex representation V of a finite group, let
$\overline {V}$
denote the mod p reduction of an integral model of V. This is only well defined up to semisimplification.
Remark 2.16 It is well known that the mod p reduction
$\overline {\mathrm {St}}$
of the (complex) Steinberg representation
$\mathrm {St}$
of
${G_p}$
is irreducible. So, it is independent of the choice of an integral model.
Lemma 2.17 Let
$\chi $
be a mod p character of
${B_p}$
. Then,
Proof See Remark 1 of [Reference Ghate and Jana6].
Theorem 2.18 Let
$0\leq r<q-1$
. Then, for the character
$\chi _r$
of
${B_q}$
, we have
-
(1)
$$ \begin{align*}\displaystyle{{\left.\mathrm{ind}_{{B_q}}^{{G_q}}\chi_r\right|}_{{G_p}}\simeq\left\{\begin{array}{@{}ll@{}} \mathrm{ind}_{{B_p}}^{{G_p}}\chi_{r}\oplus \frac{p^{f-1}-1}{p^2-1} \left( \bigoplus\limits_{i=1}^{p-1} \mathrm{ind}_{B_p}^{G_p}\chi_{i,(r-i)}\otimes\overline{\mathrm{St}} \right), & \text{ if }2\nmid f, \\ \mathrm{ind}_{{B_p}}^{{G_p}}\chi_{r}\oplus \mathrm{ind}_{T_p}^{{G_p}}\omega_{2}^{r}\oplus \frac{p\left(p^{f-2}-1\right)}{p^2-1} \left( \bigoplus \limits_{i=1}^{p-1}\mathrm{ind}_{B_p}^{G_p} \chi_{i,(r-i)} \otimes\overline{\mathrm{St}} \right), & \text{ if }2\mid f. \end{array}\right.}\end{align*} $$
-
(2)
$$ \begin{align*}\displaystyle{{\left.\mathrm{ind}_{{B_q}}^{{G_q}}\chi_r\right|}_{{G_p}}\simeq\left\{\begin{array}{@{}ll@{}} \mathrm{ind}_{{B_p}}^{{G_p}}\chi_{r}\oplus \frac{p^{f-1}-1}{p^2-1} \left(\bigoplus\limits_{i=0}^{p} \mathrm{ind}_{T_p}^{G_p}\omega_{2}^{r+i(p-1)}\right), & \text{ if }2\nmid f, \\ \mathrm{ind}_{{B_p}}^{{G_p}}\chi_{r}\oplus \mathrm{ind}_{T_p}^{{G_p}}\omega_{2}^{r} \oplus \frac{p\left(p^{f-2}-1\right)}{p^2-1} \left(\bigoplus\limits_{i=0}^{p} \mathrm{ind}_{T_p}^{G_p}\omega_{2}^{r+i(p-1)}\right), & \text{ if }2\mid f. \end{array}\right.}\end{align*} $$
Remark 2.19 Of the two decompositions of the restriction to
$G_p$
of the principal series representation of
$G_q$
given in the theorem, part (1), respectively, part (2) maximizes the number of principal series representations of
$G_p$
, respectively, induced representations from an anisotropic torus of
$G_p$
. As the reader will see, these decompositions are arrived at by a bifurcation in the proof of the theorem, rather than by deducing one from the other. A similar remark applies to the two decompositions in Theorem 3.9 below.
Proof Replace
$\chi $
by
$\chi _r$
in the isomorphism in (2.6). Write
${\chi _r|}_{{B_p}}=\chi _r$
. Using Lemma 2.14, when f is even, we get
Note that
$\left |{\mathcal {I}_1}\right |=\frac {p^{f-1}-1}{p^2-1}\text { and } \left |{\mathcal {I}_2}\right |=\frac {p\left (p^{f-2}-1\right )}{p^2-1}$
by Proposition 2.8.
By the transitivity of induction, we have
We investigate each of these isomorphisms separately.
-
(1) Note that $\text {ind}_{Z_p}^{S_p}\left ({\chi _r|}_{Z_p}\right )$
is a sum of
$p-1$
characters as
$S_p$
is abelian of order prime to p. For a character
$\psi $
of
$S_p$
, by Frobenius reciprocity,
$\psi \hookrightarrow \text {ind}_{Z_p}^{S_p}\left ({\chi _r|}_{Z_p}\right )$
if and only if
${\psi |}_{Z_p}={\chi _r|}_{Z_p}$
. Now, for all
$i\in \{1,2,\ldots , p-1\}$
, one can easily verify that
${\chi _r|}_{Z_p}={\left .\chi _{i,(r-i)}\right |}_{Z_p}$
, and
$\chi _{i,(r-i)}$
are (distinct) characters of
$S_p$
. So we get
$\text {ind}_{Z_p}^{S_p}\left ({\chi _r|}_{Z_p}\right )\simeq \bigoplus \limits _{i=1}^{p-1}\chi _{i,(r-i)}$
. So $$ \begin{align*} \text{ind}_{Z_p}^{G_p}\left({\chi_r|}_{Z_p}\right)&\simeq\text{ind}_{S_p}^{{G_p}}\left(\bigoplus\limits_{i=1}^{p-1}\chi_{i,(r-i)}\right) \\ &\simeq\bigoplus\limits_{i=1}^{p-1}\text{ind}_{S_p}^{{G_p}}\chi_{i,(r-i)}\\ &\simeq\bigoplus\limits_{i=1}^{p-1}\left(\text{ind}_{{B_p}}^{{G_p}}\chi_{i,(r-i)}\otimes\overline{\text{St}}\right)~~(\text{by Lemma}\ {2.17}). \end{align*} $$
-
(2) Similarly, $\text {ind}_{Z_p}^{T_p}\left ({\chi _r|}_{Z_p}\right )$
is a sum of
$p+1$
characters. Again, for any character
$\psi $
of
$T_p$
, we have
$\psi \hookrightarrow \text {ind}_{Z_p}^{T_p}\left ({\chi _r|}_{Z_p}\right )$
if and only if
${\psi |}_{Z_p}={\chi _r|}_{Z_p}$
. Here, one can easily verify that
${\chi _r|}_{Z_p}={\left .\omega _{2}^{r+i(p-1)}\right |}_{Z_p}$
, for all
$i\in \{0,1,\ldots , p\}$
, and
$\omega _{2}^{r+i(p-1)}$
are (distinct) characters of
$T_p$
. So we get
$\text {ind}_{Z_p}^{T_p}\left ({\chi _r|}_{Z_p}\right )\simeq \bigoplus \limits _{i=0}^{p}\omega _{2}^{r+i(p-1)}$
. So $$ \begin{align*} \text{ind}_{Z_p}^{G_p}\left({\chi_r|}_{Z_p}\right)&\simeq\text{ind}_{T_p}^{{G_p}}\left(\bigoplus\limits_{i=0}^{p}\omega_{2}^{r+i(p-1)}\right) \\&\simeq \bigoplus\limits_{i=0}^{p}\text{ind}_{T_p}^{{G_p}}\omega_{2}^{r+i(p-1)}. \\[-42pt]\end{align*} $$
3 Restriction in the cuspidal case
We know that for a non self-conjugate complex character
$\chi $
of
$T_q$
, one has
where
$\Theta (\chi )$
is the complex cuspidal representation of
${G_q}$
associated with
$\chi $
. Fixing an integral structure on each of the terms on the right induces an integral structure on the left. Taking the mod p reduction of both sides of (3.1), we see that the mod p reduction
$\overline {\Theta (\chi )}$
appears as a factor in the mod p reduction
$\overline {\mathrm {ind}_{T_q}^{{G_q}}\chi }$
. The former representation depends on the choice of the integral model of
$\Theta (\chi )$
. This is not easily written down (one approach involves crystalline cohomology [Reference Reduzzi11]). However, since
$\chi $
takes integral values, the reduction in the latter representation can be taken with respect to the natural induced integral structure, and can be identified with
$\mathrm {ind}_{T_q}^{{G_q}}\overline {\chi }$
. It therefore seems more difficult to study the restriction of the former representation to
${G_p}$
, so instead we study the restriction of the latter representation to
$G_p$
. In any case, this is consistent with our theme of studying the restrictions of induced representations.
As in the principal series case (cf. Lemma 2.1), one might expect that an obvious induction from the anisotropic torus
$T_p$
of
${G_p}$
is contained in the restriction to
${G_p}$
of the induction of a mod p character
$\chi $
from the anisotropic torus
$T_q$
of
${G_q}$
:
By Frobenius reciprocity, this is equivalent to checking if
${\chi |}_{T_p}\hookrightarrow {\left .\text {ind}_{T_q}^{{G_q}}{\chi }\right |}_{T_p}$
. This follows whenever
$T_p\subseteq T_q$
since
${\chi }\hookrightarrow {\left .\text {ind}_{T_q}^{{G_q}}{\chi }\right |}_{T_q}$
as representations of
$T_q$
(again by Frobenius reciprocity) and restriction to
$T_p$
here makes sense. However, the containment
$T_p \subseteq T_q$
is only possible if f is odd (and
$\epsilon $
is taken to be
$\eta $
, see Lemma 3.1). Indeed, if f is even and
$\left ( \begin {smallmatrix} 0 & 1 \\ \eta ^2 & 0 \end {smallmatrix} \right )$
lies in
$T_q$
, then
$\epsilon ^2 = \eta ^2$
so that
$\epsilon = \pm \eta $
lies in
${\mathbb {F}_{p^2}} \subseteq {\mathbb {F}_q}$
, which is a contradiction. See also Remark 3.6. Nevertheless, we shall prove in the second part of Theorem 3.9 the somewhat surprising fact that irrespective of the parity of f, one always has
even though
${\left .\omega _{2f}^r\right |}_{T_p}=\omega _2^r$
(only) when
$T_p\subseteq T_q$
.
In any case, it is of interest to find all the terms in the decomposition of the restriction which we proceed to do now.
3.1 The action of
${G_p}$
on
${\mathbb {P}^1({\mathbb {F}_{q^2}})}$
The action of
${G_q}$
on
${\mathbb {P}^1({\mathbb {F}_{q^2}})}$
has two orbits
${O_{{G_q}}}({\widehat {0}})$
and
${O_{{G_q}}}({\widehat {\epsilon }})$
(cf. Lemma 2.4). Also, the orbits of the
${G_p}$
action on
${\mathbb {P}^1({\mathbb {F}_{q^2}})}=\mathbb {P}^1({\mathbb F}_{p^{2f}})$
are given by (2.2). Hence,
where
${\mathcal {I}_2}$
forms a complete set of representatives of the
${G_p}\text {-orbits}$
in
$\mathbb {P}^1({\mathbb F}_{p^{2f}})\setminus {\mathbb {P}^1({\mathbb {F}_{p^2}})}$
and
$\left |{\mathcal {I}_2}\right |=\frac {p\left (p^{2f-2}-1\right )}{p^2-1}$
.
Lemma 3.1 Let
$q = p^f$
with f odd. Then,
$x\in {\mathbb {F}_{p^2}}\setminus {\mathbb {F}_p}\text { with }x^2\in {\mathbb {F}_p}$
implies that
$x\in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_q}\text { with }x^2\in {\mathbb {F}_q}$
.
Proof Let f be odd and
$x\in {\mathbb {F}_{p^2}}\setminus {\mathbb {F}_p}\text { with }x^2\in {\mathbb {F}_p}$
. Then,
$x\in {\mathbb {F}_{q^2}}$
and
$x^2\in {\mathbb {F}_q}$
. It remains to show that
$x\notin {\mathbb {F}_q}$
. If not, then
$x \in {\mathbb {F}_{p^2}} \cap {\mathbb {F}_q} = {\mathbb {F}_p}$
, a contradiction.
Notation 3.2 In view of Lemma 3.1, we can and do assume that
$\epsilon = \eta $
when f is odd.
Lemma 3.3 With respect to the above notation,
Proof Suppose
$2\mid f$
. Then,
${\mathbb {P}^1({\mathbb {F}_{p^2}})}\subseteq {\mathbb {P}^1({\mathbb {F}_q})}$
and so by comparing (3.2) and (3.3),
${O_{{G_p}}}({\widehat {\eta }})\subseteq {O_{{G_q}}}({\widehat {0}})$
. Suppose
$2\nmid f$
. Then, taking
$\epsilon =\eta $
, clearly
${O_{{G_p}}}({\widehat {\eta }})\subseteq {O_{{G_q}}}({\widehat {\epsilon }})$
.
Proposition 3.4 Let
$\epsilon \in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_q}$
such that
$\epsilon ^2\in {\mathbb {F}_q}$
. Then, the decomposition of
${O_{{G_q}}}({\widehat {\epsilon }})$
into its
${G_p}\text {-orbits}$
is given by
where
${\mathcal {J}}$
and
${\mathcal {J}'}\sqcup \{{\widehat {\epsilon }}\}$
form a complete set of representatives of
${G_p}\text {-orbits}\text { in }{O_{{G_q}}}({\widehat {\epsilon }})={\mathbb {P}^1({\mathbb {F}_{q^2}})}\setminus {\mathbb {P}^1({\mathbb {F}_q})}$
when f is even and odd, respectively. Also,
$\left |{\mathcal {J}}\right |=\frac {p^{f-1}\left (p^f-1\right )}{p^2-1}$
and
$\left |{\mathcal {J}'}\right |=\frac {\left (p^{f-1}-1\right )\left (p^f+p-1\right )}{p^2-1}$
.
Proof It is obvious that
${O_{{G_p}}}({\widehat {0}})\subseteq {O_{{G_q}}}({\widehat {0}})$
. By Equations (3.2) and (3.3),
${O_{{G_q}}}({\widehat {\epsilon }})$
is a (disjoint) union of some
${G_p}\text {-orbits}$
of
${\widehat {x}}\in {\mathcal {I}_2}$
and possibly
${O_{{G_p}}}({\widehat {\eta }})$
. In fact, by Lemma 3.3,
where we recall that
$\eta =\epsilon $
when f is odd and,
${\mathcal {J}}$
and
${\mathcal {J}'}$
are as described in the hypothesis. Their cardinalities can be computed as follows. Note that
$\left |{O_{{G_q}}}({\widehat {\epsilon }})\right |= q^2-q$
, and when f is odd,
$\left |{O_{{G_p}}}({\widehat {\epsilon }})\right |=p^2-p$
since
$\epsilon = \eta $
. Also,
$\left | {O_{{G_p}}}({\widehat {x}})\right |=p(p^2-1)$
for all
${\widehat {x}}\in {\mathcal {I}_2}$
, as can be seen in the proof of Proposition 2.8 and hence for all
${{\widehat {x}}\in {\mathcal {J}},{\mathcal {J}'}\subseteq {\mathcal {I}_2}}$
. Thus,
Remark 3.5 To summarize, the
${G_p}\text {-orbits}$
of
${\mathbb {P}^1({\mathbb {F}_{q^2}})}$
that appear in
${O_{{G_q}}}({\widehat {0}})$
and
${O_{{G_q}}}({\widehat {\epsilon }}),$
respectively, are given as follows:
where
${\widehat {\epsilon }} = {\widehat {\eta }}$
if
$2 \nmid f$
.
3.2 Decomposition of the restriction
For a character
$\chi $
of
$T_q$
, by Mackey’s restriction formula, we have
where
${T_q^\gamma }=\gamma T_q \gamma ^{-1}\cap {G_p}$
and
${\chi ^\gamma }(t)=\chi \left (\gamma ^{-1}t \gamma \right )$
for all
$t\in {T_q^\gamma }$
and where
$\Gamma '$
is a set of double coset representatives in
${G_p}\backslash {G_q}/T_q$
.
Remark 3.6 For
$\gamma = \mathrm {Id} \in \Gamma '$
, we have
${T_q^\gamma }=T_q\cap {G_p}=\left \{\begin {array}{@{}cc} Z_p, & \text { if }2\mid f, \\ T_p, & \text { if }2\nmid f. \end {array}\right .$
As a consequence,
$T_p\subseteq T_q$
if and only if f is odd.
Proof Clearly,
$Z_p\subseteq T_q\cap {G_p}$
always. Suppose f is even then we show that
${T_q\cap {G_p}\subseteq Z_p}$
. Let
$g=\left (\begin {matrix} a & b \\ b\epsilon ^2 & a \end {matrix}\right )\in T_q\cap {G_p}$
. Then,
$a,b,b\epsilon ^2\in {\mathbb {F}_p}$
and
$\epsilon \in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_q}$
. Suppose, if possible, that
$b\neq 0$
. Then,
$\epsilon ^2\in {\mathbb {F}_p}$
. This implies that
$\epsilon ^p=\pm \epsilon $
and consequently
$\epsilon ^q=\epsilon $
since f is even. This is a contradiction as
$\epsilon \notin {\mathbb {F}_q}$
. So,
$b=0$
and
$g\in Z_p$
, as desired. In particular, if
$T_p \subseteq T_q$
, then
$T_p \subseteq T_q \cap {G_p} = Z_p$
, a contradiction.
Suppose f is odd. If
$g=\left (\begin {matrix} a & b \\ b\epsilon ^2 & a \end {matrix}\right )\in T_q\cap {G_p}$
, then again
$a,b,b\epsilon ^2\in {\mathbb {F}_p}$
, but since f is odd,
$\epsilon = \eta \in {\mathbb {F}_{p^2}} \setminus {\mathbb {F}_p}$
. So g is in
$T_p$
. So
$T_q\cap {G_p} = T_p$
.
Notation 3.7 For
$a,b\in {\mathbb {F}_q}$
with
$b\neq 0$
, define 
.
Remark 3.8 Recall that
$\mathrm {Stab}_{{G_q}}({\widehat {\epsilon }})=T_q$
and
${G_q}/T_q\simeq {O_{{G_q}}}({\widehat {\epsilon }})\text { via }T_q\mapsto {\widehat {\epsilon }}$
, by Lemma 2.4. Then, a complete set of coset representatives of
${G_q}/T_q$
is given by
$\{g_{a,b}:a,b\in {\mathbb {F}_q},b\neq 0\}$
. Indeed,
and since
$T_q\mapsto {\widehat {\epsilon }}$
, we get
$g_{a,b}T_q\mapsto g_{a,b}\cdot {\widehat {\epsilon }}$
.
Under the action of
${G_p}$
on
${G_q}/T_q$
, we have
${\text {Stab}_{{G_p}}}(gT_q)=gT_q g^{-1}\cap {G_p} = T_q^g$
. Hence, by taking
$g=g_{a,b}$
and using Remark 3.8, we have
Now, the double cosets in
${G_p}\backslash {G_q}/T_q$
are in one-to-one correspondence with the orbits in
${G_p}\backslash \left ({G_q}/T_q\right )$
and hence with the orbits in
${G_p}\backslash {O_{{G_q}}}({\widehat {\epsilon }})$
by Remark 3.8. Also,
Thus, if
$\Gamma '$
is a set of double coset representatives of
${G_p}\backslash {G_q}/T_q$
, then we may assume that
$\Gamma '\subseteq \{g_{a,b}:a,b\in {\mathbb {F}_q}, b\neq 0\}$
and again by Remark 3.8,
Thus, substituting (3.5) and (3.6) in (3.4), we get
Now, Proposition 3.4 gives the description of all the
${G_p}\text {-orbits}$
in
${O_{{G_q}}}({\widehat {\epsilon }})$
and therefore,
Note that for
${\widehat {x}}\in {\mathcal {J}},{\mathcal {J}'}\subseteq {\mathcal {I}_2}$
, we have
$x\in {\mathbb {F}_{q^2}}\setminus {\mathbb {F}_{p^2}}$
by the definition of
${\mathcal {I}_2}$
, and for such
$x, ~{\text {Stab}_{{G_p}}}({\widehat {x}})=Z_p$
by Lemma 2.7. Thus,
${\text {Stab}_{{G_p}}}(\widehat {a+b\epsilon })=Z_p$
, for all
$\widehat {a+b\epsilon }\in {\mathcal {J}}$
and
${\mathcal {J}'}$
. In this case,
${\left .\chi ^{g_{a,b}}\right |}_{Z_p}={\chi |}_{Z_p}$
. Also, for f odd,
$\epsilon =\eta $
and so
${\text {Stab}_{{G_p}}}({\widehat {\epsilon }})=T_p$
. Note that
$g_{0,1}=\mathrm {Id}$
and so
$\chi ^{g_{0,1}}={\chi |}_{T_p}$
. Combining this information, we get
Theorem 3.9 Let
$0\leq r<q^2-1$
. Then, for the character
$\omega _{2f}^r$
of
$T_q$
, we have
-
(1)
$$ \begin{align*}\!\!\!\displaystyle{{\left.\mathrm{ind}_{T_q}^{{G_q}}\omega_{2f}^r\right|}_{{G_p}}\simeq\left\{\begin{array}{@{}ll@{}} \frac{p^{f-1}\left(p^f-1\right)}{p^2-1}\left(\bigoplus\limits_{i=1}^{p-1}\mathrm{ind}_{B_p}^{G_p} \chi_{i,(r-i)}\otimes\overline{\mathrm{St}}\right), & \text{ if }2\mid f, \\ \mathrm{ind}_{T_p}^{G_p}\omega_2^r~\oplus \frac{\left(p^{f-1}-1\right)\left(p^f+p-1\right)}{p^2-1}\left(\bigoplus\limits_{i=1}^{p-1}\mathrm{ind}_{B_p}^{G_p} \chi_{i,(r-i)}\otimes\overline{\mathrm{St}}\right), & \text{ if }2\nmid f. \end{array}\right.}\end{align*} $$
-
(2)
$$ \begin{align*}\displaystyle{{\left.\mathrm{ind}_{T_q}^{{G_q}}\omega_{2f}^r\right|}_{{G_p}}\simeq\left\{\begin{array}{@{}ll@{}} \frac{p^{f-1}\left(p^f-1\right)}{p^2-1}\left(\bigoplus\limits_{i=0}^{p} \mathrm{ind}_{T_p}^{G_p}\omega_{2}^{r+i(p-1)}\right), & \text{ if }2\mid f, \\ \mathrm{ind}_{T_p}^{G_p}\omega_2^r~\oplus \frac{\left(p^{f-1}-1\right)\left(p^f+p-1\right)}{p^2-1}\left(\bigoplus\limits_{i=0}^{p} \mathrm{ind}_{T_p}^{G_p}\omega_{2}^{r+i(p-1)}\right), & \text{ if }2\nmid f. \end{array}\right.}\end{align*} $$
Proof Replace
$\chi $
by
$\omega _{2f}^r$
in the isomorphism in (3.7). Then,
${\left .\omega _{2f}^r\right |}_{T_p}=\omega _2^r$
as characters of
$T_p$
if f is odd. Also,
$\left |{\mathcal {J}}\right |=\frac {p^{f-1}\left (p^f-1\right )}{p^2-1}\text { and } \left |{\mathcal {J}'}\right |=\frac {\left (p^{f-1}-1\right )\left (p^f+p-1\right )}{p^2-1}$
by Proposition 3.4. By the transitivity of induction, we have
We investigate each of these isomorphisms separately.
-
(1) Note that $\text {ind}_{Z_p}^{S_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )$
is a sum of
$p-1$
characters as
$S_p$
is abelian of order prime to p. For a character
$\psi $
of
$S_p$
, by Frobenius reciprocity,
$\psi \hookrightarrow \text {ind}_{Z_p}^{S_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )$
if and only if
${\psi |}_{Z_p}={\left .\omega _{2f}^r\right |}_{Z_p}$
. Now, for all
$i\in \{1,2,\ldots , p-1\}$
, one can easily verify that
${\left .\omega _{2f}^r\right |}_{Z_p}={\left .\chi _{i,(r-i)}\right |}_{Z_p}$
, where
$\chi _{i,(r-i)}$
are (distinct) characters of
$S_p$
. So we get
$\text {ind}_{Z_p}^{S_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )\simeq \bigoplus \limits _{i=1}^{p-1}\chi _{i,(r-i)}$
. So, $$ \begin{align*} \text{ind}_{Z_p}^{G_p}\left({\left.\omega_{2f}^r\right|}_{Z_p}\right)&\simeq\text{ind}_{S_p}^{{G_p}}\left(\bigoplus\limits_{i=1}^{p-1}\chi_{i,(r-i)}\right) \\ &\simeq \bigoplus\limits_{i=1}^{p-1}\left(\text{ind}_{S_p}^{{G_p}}\chi_{i,(r-i)}\right)\\ &\simeq\bigoplus\limits_{i=1}^{p-1}\left(\text{ind}_{{B_p}}^{{G_p}}\chi_{i,(r-i)}\otimes\overline{\text{St}}\right)~~( \text{by Lemma}\ {2.17}). \end{align*} $$
-
(2) Similarly, $\text {ind}_{Z_p}^{T_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )$
is a sum of
$p+1$
characters. Again, for any character
$\psi $
of
$T_p$
, we have
$\psi \hookrightarrow \text {ind}_{Z_p}^{T_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )$
if and only if
${\psi |}_{Z_p}={\left .\omega _{2f}^r\right |}_{Z_p}$
. Here, one can easily verify that
${\left .\omega _{2f}^r\right |}_{Z_p}={\left .\omega _{2}^{r+i(p-1)}\right |}_{Z_p}$
, for all
$i\in \{0,1,\ldots , p\}$
, where
$\omega _{2}^{r+i(p-1)}$
are (distinct) characters of
$T_p$
. So we get
$\text {ind}_{Z_p}^{T_p}\left ({\left .\omega _{2f}^r\right |}_{Z_p}\right )\simeq \bigoplus \limits _{i=0}^{p}\omega _{2}^{r+i(p-1)}$
. Thus, $$ \begin{align*} \text{ind}_{Z_p}^{G_p}\left({\left.\omega_{2f}^r\right|}_{Z_p}\right)&\simeq\text{ind}_{T_p}^{{G_p}}\left(\bigoplus\limits_{i=0}^{p}\omega_{2}^{r+i(p-1)}\right) \\&\simeq \bigoplus\limits_{i=0}^{p}\text{ind}_{T_p}^{{G_p}}\omega_{2}^{r+i(p-1)}. \end{align*} $$
Remark 3.10 Consider
$\omega _{2}$
both as a character of
$T_p$
taking values in a mod p field and in a characteristic zero field by taking its Teichmüller lift. Then, for
$0\leq r<p^2-1$
with
$\omega _{2}^{r}$
not self-conjugate, by using the mod p reduction of the isomorphism in (3.1) with
$\chi =\omega _2^r$
, we can further substitute
$\overline {\Theta \left (\omega _{2}^{r}\right )}\otimes \overline {\mathrm {St}}$
for
$\mathrm {ind}_{T_p}^{{G_p}}\omega _{2}^{r}$
in the statements of Theorems 2.18 and 3.9. We also remark that the mod p reductions of complex cuspidal representations of
$G_p$
have been studied in, for instance, [Reference Ghate and Jana6, Reference Reduzzi11, Reference Schein13].
Remark 3.11 Relative versions of the results in this article studying the restrictions of mod p representations of
$\mathrm {GL}_2({{\mathbb F}_{q'}})$
of the type considered in this article to
$\mathrm {GL}_2({\mathbb {F}_q})$
for
$q'$
a power of q could easily be further derived. For brevity, we have decided to treat only the absolute case
$q = p$
.
Acknowledgements
The second author thanks Prof. Eknath Ghate for suggesting the problem, for his invitation to visit TIFR, Mumbai and for the warm hospitality extended during her visit and thanks Prof. Gautam Borisagar for his valuable inputs. She is also grateful to CSIR for the Ph.D. fellowship. Both authors thank the referee for several suggestions that have improved the article.
