1 Notation and conventions
Let
$p$
be a prime. Let
$\omega $
and
$\omega _2$
be the fundamental characters of
$I_{\mathbb {Q}_p}$
of levels
$1$
and
$2$
. The character
$\omega $
has a canonical extension to
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
. For an integer
$c$
with
$p + 1 \nmid c$
, we choose an extension of
$\omega _2^c$
to
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_{p^2})$
so that the irreducible representation
$\mathrm {ind}(\omega _2^c)$
obtained by inducing this extension from
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_{p^2})$
to
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
has determinant
$\omega ^c$
. Let
$\mu _{\lambda }$
be the unramified character of
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
sending a geometric Frobenius to
$\lambda \in \overline {\mathbb {F}}_p^*$
.
Let
$G$
be the group
$\mathrm {GL}_2(\mathbb {Q}_p)$
and
$Z$
be its center. Let
$K$
be the maximal compact subgroup
$\mathrm {GL}_2(\mathbb {Z}_p)$
. The Iwahori subgroup of upper triangular matrices mod
$p$
is denoted by
$I$
.
Let
$$\begin{align*}\alpha = \begin{pmatrix}1 & 0 \\ 0 & p\end{pmatrix}, \beta = \begin{pmatrix}0 & 1 \\ p & 0\end{pmatrix} \text{ and } w = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}. \end{align*}$$
For
$r \geq 0$
, let
$V_r$
be the symmetric
$r$
-th power of the standard representation
$\mathbb {F}_p^2$
of
$K$
. Let
$V_r^* \subset V_r$
be the
$K$
-subrepresentation of polynomials divisible by
$X^pY - XY^p$
. We think of these as representations of
$KZ$
by making the scalar matrix
$p$
act trivially. Let
$d^r$
be the character
$I \to \mathbb {F}_p^*$
that sends
$\begin {pmatrix}a & b \\ c & d\end {pmatrix} \in I$
to
$d^r \ \mod p$
and extend it to
$IZ$
by sending
$p \in Z$
to
$1$
.
Let
$\mathrm {ind}_{KZ}^G\> \sigma $
denote the compact induction of a representation
$\sigma $
of
$KZ$
over
$\mathbb {F}_p$
. Functions in this space have compact support modulo
$KZ$
and are therefore finite linear combinations of functions of the form
$[g, v]$
for
$g \in G$
and
$v \in \sigma $
, where
$$\begin{align*}[g, v](g') = \begin{cases} g'gv, & \text{ if } g' \in KZg^{-1}, \\ 0, & \text{ if } g' \not \in KZg^{-1}. \end{cases} \end{align*}$$
Similarly, for a character
$\chi $
of
$IZ$
, let
$\mathrm {ind}_{IZ}^G\> \chi $
denote the compact induction of
$\chi $
and let
$\mathrm {ind}_{IZ}^{KZ}\> \chi $
denote the usual induction of
$\chi $
. Functions in these spaces are finite linear combinations of functions of the form 
and
$[\kern-2.6pt[\kern-2.6pt[g, v]\kern-2.6pt]\kern-2.6pt]$
, respectively, where 
and
$[\kern-2.6pt[\kern-2.6pt[g, v]\kern-2.6pt]\kern-2.6pt]$
are defined analogous to
$[g, v]$
. By spherical Hecke algebra, we mean the algebra
$\mathrm {End}_G(\mathrm {ind}_{KZ}^G\> \sigma )$
for an irreducible representation
$\sigma $
of
$KZ$
. Similarly, by Iwahori–Hecke algebra, we mean
$\mathrm {End}_G(\mathrm {ind}_{IZ}^G\> d^r)$
for
$0 \leq r \leq p - 1$
.
Let
$[a] \in \{0, \ldots , p - 2\}$
be the class of an integer
$a$
mod
$p - 1$
. Let
$\delta _{a, b} = 1$
if
$a = b$
and
$0$
otherwise. Let
$I_1$
be the set consisting of
$(p-1)\text{-}$
th roots of unity in
$\mathbb {Z}_p$
and
$0$
.
2 Introduction
Let
$F$
be a local field of residue characteristic
$p$
. The study of smooth irreducible representations of
$\mathrm {GL}_2(F)$
with a central character over fields of characteristic
$p$
was initiated by Barthel and Livné in [Reference Barthel and LivnéBL95] and [Reference Barthel and LivnéBL94]. In [Reference BreuilBre03a], Breuil studied the particularly tricky sub-class of such representations called the supersingular representations and proved that they are irreducible when
$F = \mathbb {Q}_p$
. He then wrote down the following definition, which he called the mod
$p$
local Langlands correspondence (mod
$p$
LLC), which is an injection from isomorphism classes of two-dimensional semi-simple mod
$p$
representations of
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
to isomorphism classes of smooth representations of
$\mathrm {GL}_2(\mathbb {Q}_p)$
.
Definition 2.1 [Reference BreuilBre03a, Definition 4.2.4]
For
$r \in \{0, \ldots , p - 1\}$
,
$\lambda \in \overline {\mathbb {F}}_p$
and a smooth character
$\eta : \mathbb {Q}_p^* \to \overline {\mathbb {F}}_p^*$
, define
-
(i) If
$\lambda = 0$
: -
(ii) If
$\lambda \neq 0$
:
More precisely, this is the semi-simple version of the mod
$p$
LLC, which is all we consider in this paper. There is also a version without semi-simplification by Breuil [Reference BreuilBre10, Definition 2.2] and a functorial version due to Colmez [Reference ColmezCol10].
One of the applications of this correspondence is to compute the reduction mod
$p$
of irreducible two-dimensional crystalline representations of
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
. Let us explain this idea now. For each
$k \geq 2$
and
$a_p \in \overline {\mathbb {Q}}_p$
of positive valuation, there exists an irreducible, two-dimensional crystalline representation
$V_{k, a_p}$
of
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
over a
$p$
-adic number field. Let
$B(V_{k, a_p})$
be the Banach space associated with
$V_{k, a_p}$
under the
$p$
-adic local Langlands correspondence. In [Reference BreuilBre03b], Breuil conjectured that (the semi-simplification of) the reduction mod
$p$
of
$V_{k, a_p}$
corresponds to (the semi-simplification of) the reduction mod
$p$
of
$B(V_{k, a_p})$
under the bijection given in Definition 2.1. Using this conjecture, he computed the reduction mod
$p$
of
$V_{k, a_p}$
for
$k \leq 2p$
$(k \neq 4$
if
$p = 2$
) and all
$a_p$
. The following famous diagram illustrates this idea:
Breuil’s conjecture was subsequently verified by Berger in [Reference BergerBer10].
This idea of using the compatibility with respect to reduction mod
$p$
between the
$p$
-adic and mod
$p$
local Langlands correspondences was used in [Reference Buzzard and GeeBG09, Reference Buzzard and GeeBG13, Reference Ganguli and GhateGG15, Reference Bhattacharya and GhateBG15, Reference Bhattacharya, Ghate and RozensztajnBGR18, Reference Ghate and RaiGR23] to compute the reduction of
$V_{k, a_p}$
for unbounded weights
$k$
but bounded slopes
$v_p(a_p)$
. Thanks to the results in these papers, the reduction of
$V_{k, a_p}$
is known for arbitrary
$k \geq 2$
,
$0 < v_p(a_p) < 2$
and
$p$
odd (and in some cases
$p \geq 5$
). Nagel–Pande [Reference Nagel and PandeNP19] and Arsovski [Reference ArsovskiArs21] also have some results for higher slopes. The growing complications in all these results were the reason why Breuil restricted to
$k \leq 2p$
(see the comments after [Reference BreuilBre03b, Théorème 1.4]). The phenomenon of growing complications can be seen in [Reference GhateGha21], in the zig-zag conjecture (now a theorem due to [Reference GhateGha23]) of Ghate, which describes the reduction mod
$p$
of
$V_{k, a_p}$
in one of the trickiest cases, namely for weights
$k \equiv 2v_p(a_p) + 2\ \mod p - 1$
and slopes
$v_p(a_p)$
in the interval
$\left(0, \frac {p - 1}{2}\right]$
. Recently Bhattacharya, Ghate, and Vangala have computed
$\overline {V}_{k, a_p}$
in many cases for arbitrary
$k \geq 2$
and
$0 < v_p(a_p) < p$
.
In this paper, we rewrite Breuil’s mod
$p$
LLC using the language of Iwahori induction:
Theorem 2.2 (Iwahori mod
$p$
LLC)
For
$r \in \{0, \ldots , p - 1\}$
,
$\lambda \in \overline {\mathbb {F}}_p$
and a smooth character
$\eta : \mathbb {Q}_p^* \to \overline {\mathbb {F}}_p^*$
, define
-
• If
$\lambda = 0:$
-
• If
$\lambda \neq 0:$
$$ \begin{align*} \!\!(\mu_{\lambda}\omega^{r + 1} \oplus \mu_{\lambda^{-1}})\otimes \eta \leftrightarrow &\left(\frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0} + \delta_{r, p - 1}T_{1, 0}) + (T_{1, 2} + \delta_{r, 0}T_{1, 0} - \lambda)} \otimes \eta \!\right)^{\!\!\mathrm{ss}} \oplus \\ & \left(\frac{\mathrm{ind}_{IZ}^G\> d^{[p - 3 - r]}}{(T_{-1, 0} + \delta_{[p - 3 - r], p - 1}T_{1, 0}) + (T_{1, 2} + \delta_{[p - 3 - r], 0}T_{1, 0} - \lambda^{-1})}\otimes \eta\omega^{r + 1}\!\right)^{\!\!\mathrm{ss}}. \end{align*} $$
One of the applications of the Iwahori mod
$p$
LLC is to compute the reduction of irreducible, two-dimensional semi-stable non-crystalline representations of
$\mathrm {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$
. The method to compute these reductions is analogous to the one discussed above for crystalline representations. Namely, the reduction of such a semi-stable representation is the one that corresponds, under Theorem 2.2, to the reduction of the Banach space associated with it by the
$p$
-adic LLC. This has been carried out recently by Ghate and the author for semi-stable representations with difference of Hodge–Tate weights lying in the range
$[2, p]$
in a preprint [Reference Chitrao and GhateCG23]. In particular, we have managed to compute the reduction mod
$p$
of the semi-stable representation
$V_{k, \mathcal {L}}$
for
$k = p$
and
$p + 1$
and arbitrary
$\mathcal {L}$
. To the best of our knowledge, this is the first time the compatibility between the
$p$
-adic and the mod
$p$
LLC has been used to compute the reduction mod
$p$
of semi-stable representations.
3 Comparing Iwahori induction and spherical induction
In this section, we recall the Hecke algebra for the Iwahori induction of the mod
$p$
character
$d^r$
. We also recall old comparison theorems and prove new ones between
$\mathrm {ind}_{KZ}^G\> V_{r}$
and certain quotients of
$\mathrm {ind}_{IZ}^G\> d^r$
. The
$0 < r < p - 1$
case was treated by Anandavardhanan and Borisagar in [Reference Anandavardhanan and BorisagarAB15]. We treat the cases
$r = 0, \> p - 1$
. This involves studying the corresponding non-commutative Iwahori–Hecke algebra.
3.1 Recall of results in the commutative Hecke algebra case
It is well known (cf. [Reference Barthel and LivnéBL95]) that if
$0 < r < p - 1$
, then the endomorphism algebra of the representation
$\mathrm {ind}_{IZ}^G\> d^r$
is the commutative Hecke algebra generated by two operators
$T_{-1, 0}$
and
$T_{1, 2}$
satisfying
$$\begin{align*}T_{-1, 0}T_{1, 2} = 0 = T_{1, 2}T_{-1, 0}. \end{align*}$$
These operators are defined by the following formulas (cf. [Reference Anandavardhanan and BorisagarAB15])
where
$v$
is a basis vector for the vector space underlying the representation
$d^r$
.
Similarly, we have a Hecke algebra for representations induced from
$KZ$
to
$G$
. The endomorphism algebra for the representation
$\mathrm {ind}_{KZ}^G\> V_r$
is generated by a single operator
$T$
. Its formula is given by
$$ \begin{align} T[\mathrm{id}, P(X, Y)] = \sum_{\lambda \in I_1}\left[\begin{pmatrix}p & \lambda \\ 0 & 1\end{pmatrix}, P(X, -\lambda X + pY)\right] + \left[\begin{pmatrix}1 & 0 \\ 0 & p\end{pmatrix}, P(pX, Y)\right]. \end{align} $$
The following results appear in [Reference Anandavardhanan and BorisagarAB15]:
Proposition 3.1 [Reference Anandavardhanan and BorisagarAB15, Proposition 3.1]
We have
$$\begin{align*}\mathrm{Im}\> T_{-1, 0} = \mathrm{Ker}\> T_{1, 2} \text{ and } \mathrm{Im}\> T_{1, 2} = \mathrm{Ker}\> T_{-1, 0}. \end{align*}$$
Theorem 3.2 [Reference Anandavardhanan and BorisagarAB15, Theorem 4.1]
Let
$0 < r < p - 1$
. We have a
$G$
-equivariant isomorphism
$$\begin{align*}\mathrm{ind}_{KZ}^G\> V_r \simeq \frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0})}, \end{align*}$$
which sends
$[\mathrm {id}, x^r]$
to 
. Under this isomorphism, the Hecke operator
$T$
is mapped to
$T_{1, 2}$
.
Theorem 3.3 [Reference Anandavardhanan and BorisagarAB15, p. 18]
For
$0 < r < p - 1$
, there is an isomorphism
$$\begin{align*}\mathrm{ind}_{IZ}^G\> d^r \simeq \mathrm{ind}_{IZ}^G\> a^r, \end{align*}$$
which sends 
to 
. Under this isomorphism,
$T_{-1, 0}$
is mapped to
$T_{1, 2}$
and vice versa.
Remark 3.4 Theorem 3.2 was generalized by Jana in [Reference JanaJan23, Theorem
$1.2$
]. Anandavardhanan and Jana later used this generalization to compute the space of pro-
$p$
-Iwahori invariant vectors of a quotient of the universal supersingular representation of
$\mathrm {GL}_2(F)$
, see [Reference Anandavardhanan and JanaAJ21, Theorem
$1.2$
].
3.2 New results in the noncommutative Hecke algebra case
If
$r = 0$
,
$p - 1$
, then the endomorphism algebra of
$\mathrm {ind}_{IZ}^G\> d^r = \mathrm {ind}_{IZ}^G\> 1$
is no longer only generated by
$T_{1, 2}$
and
$T_{-1, 0}$
. Indeed, the naïve generalization of Theorem 3.3 suggests that there is an extra endomorphism of
$\mathrm {ind}_{IZ}^G\> 1$
given by 
. In fact, this operator exists and is denoted by
$T_{1, 0}$
. It is proved in [Reference Barthel and LivnéBL95] that the corresponding Hecke algebra is non-commutative and turns out to be generated by
$T_{1, 0}$
and
$T_{1, 2}$
subject to the relations
$$ \begin{align} T_{1, 0}^2 = 1 \text{ and } T_{1, 2}T_{1, 0}T_{1, 2} = -T_{1, 2}. \end{align} $$
We write down explicit formulas for these Iwahori–Hecke operators acting on compactly supported functions on the Bruhat–Tits tree
$\Delta $
of
$\mathrm {SL}_2(\mathbb {Q}_p)$
. Recall that the vertices of
$\Delta $
correspond to homothety classes of lattices in
$\mathbb {Q}_p^2$
. There is a “canonical” vertex, namely, the one represented by
$\mathbb {Z}_p^2$
. Two vertices in
$\Delta $
are joined by an edge in
$\Delta $
if they are represented by lattices
$L$
and
$L'$
such that
$pL \subsetneq L' \subsetneq L$
. We state some simple lemmas without proof. In the next lemma, by the term “oriented edge”, we mean an ordered pair of adjacent vertices in
$\Delta $
.
Lemma 3.5 The oriented edges of
$\Delta $
are in a one-to-one correspondence with
$G/IZ$
.
We therefore identify elements in the representation
$\mathrm {ind}_{IZ}^G\> 1$
with functions on the oriented edges of
$\Delta $
with finite support: The element 
is identified with the function that maps the edge
$g (\mathbb {Z}_p^2, \alpha \mathbb {Z}_p^2)$
to
$1$
and all other edges to
$0$
. In the following lemma,
$\overline {e}$
denotes the edge
$e$
with reverse orientation.
Lemma 3.6 If
$e$
is the edge corresponding to the function 
, then
$\overline {e}$
is the edge corresponding to the function 
.
In the next lemma,
$o(e)$
and
$t(e)$
denote the origin and terminal of an edge
$e$
, respectively.
Lemma 3.7 If
$e$
is the edge corresponding to 
, then the edges
$e'$
with
$o(e') = t(e)$
and
$e' \neq \overline {e}$
correspond to 
, where
$\lambda \in I_1$
.
Using [Reference Barthel and LivnéBL95, Lemma 6] and Lemmas 3.6, 3.7, the formulas for
$T_{1, 0}$
and
$T_{1, 2}$
are given by
Remark 3.8
-
• By Frobenius reciprocity, the Iwahori–Hecke algebra
$\mathrm {End}_G(\mathrm {ind}_{IZ}^G\> 1)$
is identified with the space
$\mathrm {Hom}_{IZ}(1, \mathrm {Res}^{G}_{IZ}\mathrm {ind}_{IZ}^G\> 1)$
. Since the action of
$G$
on functions in
$\mathrm {ind}_{IZ}^G\> 1$
is by right translations, this last space is isomorphic to functions in
$\mathrm {ind}_{IZ}^G\> 1$
that are right invariant under
$IZ$
. In other words, we can identify elements in this Iwahori–Hecke algebra with functions supported on a finite number of double cosets of
$IZ$
in
$G$
. Under this identification,
$T_{1, 0}$
and
$T_{1, 2}$
correspond to and , respectively, as explained in [Reference Barthel and LivnéBL95]. -
• The formula for
$T_{1, 0}$
changes if we replace
$\mathrm {ind}_{IZ}^G\> 1$
by
$\mathrm {ind}_{IZ}^G\> \omega ^s$
for any
$0 \leq s \leq p - 2$
but the formula for
$T_{1, 2}$
remains the same. Indeed, we have (5)This follows from the projection formula which states that the map
$\mathrm {ind}_{IZ}^G\> \omega ^s \to \omega ^s \otimes (\mathrm {ind}_{IZ}^G\> 1)$
sending to is an isomorphism of
$G$
-representations.
and 
, respectively, as explained in [
to 
is an isomorphism of 
The operator
$T_{-1, 0}$
satisfies the relation (see [Reference Barthel and LivnéBL95, Lemma 8(4)])
$$ \begin{align} T_{-1, 0} = T_{1, 0}T_{1, 2}T_{1, 0}. \end{align} $$
The formula for
$T_{-1, 0}$
continues to be given by (1).
We now extend the results of [Reference Anandavardhanan and BorisagarAB15] to this non-commutative setting. We will use the following classical lemma from, e.g., [Reference Ghate and JanaGJ23] (the group
$\mathrm {B}(\mathbb {F}_p)$
denotes the group of upper triangular matrices with entries in
$\mathbb {F}_p$
):
Lemma 3.9 [Reference Ghate and JanaGJ23, Theorem 1.1]
For
$r \geq p + 1$
, the map
$$\begin{align*}\psi : V_r \to \mathrm{ind}\>_{\mathrm{B}(\mathbb{F}_p)}^{\mathrm{GL}_2(\mathbb{F}_p)} d^r \end{align*}$$
that sends a polynomial
$P \in V_r$
to the function
$\psi _P : \mathrm {GL}_2(\mathbb {F}_p) \to \mathbb {F}_p$
defined by
$$\begin{align*}\psi_P\left(\begin{pmatrix}a & b \\ c & d\end{pmatrix}\right) = P(c, d) \end{align*}$$
induces an isomorphism of
$\mathrm {GL}_2(\mathbb {F}_p)$
-representations
$$\begin{align*}V_r/V_r^* \xrightarrow{\sim} \mathrm{ind}\>_{\mathrm{B}(\mathbb{F}_p)}^{\mathrm{GL}_2(\mathbb{F}_p)} d^r. \end{align*}$$
We identify the
$KZ$
-representation obtained by inflating the
$\mathrm {GL}_2(\mathbb {F}_p)$
-representation
$\mathrm {ind}\>_{\mathrm {B}(\mathbb {F}_p)}^{\mathrm {GL}_2(\mathbb {F}_p)} d^r$
via the reduction mod
$p$
map
$K \to \mathrm {GL}_2(\mathbb {F}_p)$
(and making the scalar matrix
$p$
act trivially) with
$\mathrm {ind}_{IZ}^{KZ}\> d^r$
since the subgroup
$I$
reduces mod
$p$
to
$\mathrm {B}(\mathbb {F}_p)$
.
We first prove an analog of Proposition 3.1.
Proposition 3.10 We have
$$\begin{align*}\mathrm{Ker}\> T_{1, 2}T_{1, 0} = \mathrm{Im}\> (1 + T_{1, 2}T_{1, 0}) \text{ and } \mathrm{Ker}\> (1 + T_{1, 2}T_{1, 0}) = \mathrm{Im}\> T_{1, 2}T_{1, 0}. \end{align*}$$
Proof Note that the second identity in (3) implies that the operator
$-T_{1, 2}T_{1, 0}$
is an idempotent. This proposition then follows from the theory of idempotent endomorphisms.
The following remark is the analog of Theorem 3.3.
Remark 3.11 The map
$T_{1, 0}$
induces an isomorphism
$\mathrm {ind}_{IZ}^G\> 1 \to \mathrm {ind}_{IZ}^G\> 1$
. Under this map, the operator
$T_{1, 2}$
is mapped to
$T_{-1, 0}$
and vice versa. This is because of relation (6) and
$T_{1, 0}^2 = 1$
.
Before proving the analog of Theorem 3.2, we state and prove a lemma.
Lemma 3.12 The representation
$V_{2p - 2}/V_{2p - 2}^*$
splits as a direct sum
$V_{p - 1} \oplus V_0$
. The copies of
$V_{p - 1}$
and
$V_{0}$
in
$V_{2p - 2}/V_{2p - 2}^*$
are the subspaces generated by
$X^{2p - 2}$
and
$X^{2p - 2} - X^{p - 1}Y^{p - 1} + Y^{2p - 2}$
, respectively.
Proof By [Reference BreuilBre03b, Lemme 5.1.3(ii)], we have the following exact sequence
$$\begin{align*}0 \to V_{p - 1} \to V_{2p - 2}/V_{2p - 2}^* \to V_{0} \to 0, \end{align*}$$
which we think of as an exact sequence of
$\mathrm {GL}_2(\mathbb {F}_p)$
-representations. Since
$V_{p - 1}$
is a projective, and hence an injective
$\mathrm {GL}_2(\mathbb {F}_p)$
-representation, this sequence splits. The injection of
$V_{p - 1}$
into
$V_{2p - 2}/V_{2p - 2}^*$
is given by sending the generator
$X^{p - 1}$
to
$X^{2p - 2}$
by [Reference BreuilBre03b, Lemme 5.1.3(ii)]. Next, note that there is a unique copy of
$V_0$
in
$V_{2p - 2}/V_{2p - 2}^*$
. Using Lemma 3.9, we see that
$$ \begin{align*}V_{2p - 2}/V_{2p - 2}^* \simeq \mathrm{ind}_{\mathrm{B}(\mathbb{F}_p)}^{\mathrm{GL}_2(\mathbb{F}_p)} 1.\end{align*} $$
Clearly, the copy of
$V_0$
in
$\mathrm {ind}_{\mathrm {B}(\mathbb {F}_p)}^{\mathrm {GL}_2(\mathbb {F}_p)} 1$
is given by the constant functions. A short computation shows that under the explicit isomorphism given in Lemma 3.9, the polynomial
$X^{2p - 2} - X^{p - 1}Y^{p - 1} + Y^{2p - 2}$
maps to the constant function
$1$
.
Theorem 3.13 The map
induces isomorphisms
$$ \begin{align*} \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{1, 2}T_{1, 0})} \simeq \mathrm{ind}_{KZ}^G\> V_0\quad \text{and}\quad \frac{\mathrm{ind}_{IZ}^G\> 1}{(1 + T_{1, 2}T_{1, 0})} \simeq \mathrm{ind}_{KZ}^G\> V_{p - 1}. \end{align*} $$
Moreover, the operators
$T_{-1, 0} + T_{1, 0}$
and
$T_{-1, 0}$
on the left correspond to the operator
$T$
on the right under the isomorphisms mentioned above.
Proof We first prove that the map
is an isomorphism. Since
$IZ \subseteq KZ \subseteq G$
with
$IZ \subseteq KZ$
of finite index,
$\mathrm {ind}_{IZ}^G\> 1$
is isomorphic to
$\mathrm {ind}_{KZ}^G\> \mathrm {ind}_{IZ}^{KZ}\> 1$
by sending
where single brackets denote functions in
$\mathrm {ind}_{KZ}^G\> (\mathrm {ind}_{IZ}^{KZ}\> 1)$
, double brackets denote functions in
$\mathrm {ind}_{IZ}^G\> 1$
and triple brackets denote functions in
$\mathrm {ind}_{IZ}^{KZ}\> 1$
. A short computation using Lemma 3.9 shows that the map sending
$Y^{2p - 2} - X^{p - 1}Y^{p - 1}$
to
$[\kern-2.6pt[\kern-2.6pt[\mathrm {id}, 1]\kern-2.6pt]\kern-2.6pt]$
is an isomorphism between
$V_{2p - 2}/V_{2p - 2}^*$
and
$\mathrm {ind}_{IZ}^{KZ}\> 1$
.
Using the relation
$T_{1, 2}T_{1, 0}T_{1, 2} = -T_{1, 2}$
in the noncommutative Hecke algebra for
$\mathrm {ind}_{IZ}^G\> 1$
, we see that the operator
$-T_{1, 2}T_{1, 0}$
is an idempotent. Therefore, we get a splitting
$$\begin{align*}\mathrm{ind}_{IZ}^G\> 1 = \mathrm{Im}\> (-T_{1, 2}T_{1, 0}) \oplus \mathrm{Im}\> (1 + T_{1, 2}T_{1, 0}). \end{align*}$$
We also know from Lemma 3.12 that
$$\begin{align*}\mathrm{ind}_{KZ}^G\> V_{2p - 2}/V_{2p - 2}^* = \mathrm{ind}_{KZ}^G\> V_{p - 1} \oplus \mathrm{ind}_{KZ}^G\> V_{0}, \end{align*}$$
where
$\mathrm {ind}_{KZ}^G\> V_{p - 1}$
is identified with the subspace generated by
$[\mathrm {id}, X^{2p - 2}]$
and
$\mathrm {ind}_{KZ}^G\> V_{0}$
is identified with the subspace generated by
$[\mathrm {id}, X^{2p - 2} - X^{p - 1}Y^{p - 1} + Y^{2p - 2}]$
. Therefore, to prove the first part of this theorem, we need to show that:
-
•
$\mathrm {Im}\> T_{1, 2}T_{1, 0} \mapsto \mathrm {ind}_{KZ}^G\> V_{p - 1}$
-
•
$\mathrm {Im}\> (1 + T_{1, 2}T_{1, 0}) \mapsto \mathrm {ind}_{KZ}^G\> V_{0}.$
Indeed, under the map
$\mathrm {ind}_{IZ}^G\> 1 \to \mathrm {ind}_{KZ}^G\> V_{2p - 2}/V_{2p - 2}^*$
, the element 
maps to
$$\begin{align*}\sum_{\lambda \in I_1}\left[\begin{pmatrix}1 & 0 \\ p\lambda & p\end{pmatrix}, Y^{2p - 2} - X^{p - 1}Y^{p - 1}\right] = \sum_{\lambda \in I_1}\left[\beta\begin{pmatrix}1 & \lambda \\ 0 & 1\end{pmatrix}w, Y^{2p - 2} - X^{p - 1}Y^{p - 1}\right]. \end{align*}$$
Since
$\begin {pmatrix}1 & \lambda \\ 0 & 1\end {pmatrix}w \in KZ$
, we transfer it to the other side to get
$$\begin{align*}\sum_{\lambda \in I_1}[\beta, X^{2p - 2} - (\lambda X + Y)^{p - 1}X^{p - 1}]. \end{align*}$$
Expanding
$(\lambda X + Y)^{p - 1}$
and summing over
$\lambda $
using the identity
$$ \begin{align} \sum_{i = 0}^{p - 1}[i]^j = \begin{cases} 0 & \text{ if } p - 1 \nmid j \\ p - 1 & \text{ if } p - 1 \mid j, \end{cases} & \quad\text{for } j \geq 1, \end{align} $$
we see that
-
• Using (8), we see that
Moreover,
$[\mathrm {id}, X^{2p - 2}]$
generates
$\mathrm {ind}_{KZ}^G\> V_{p - 1}$
in
$\mathrm {ind}_{KZ}^G\> V_{2p - 2}/V_{2p - 2}^*$
.
-
• Again using (8), we see that
Moreover,
$[\mathrm {id}, X^{2p - 2} - X^{p - 1}Y^{p - 1} + Y^{2p - 2}]$
generates
$\mathrm {ind}_{KZ}^G\> V_{0}$
.
We have proved that the natural map induces isomorphisms
$$ \begin{align} \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{1, 2}T_{1, 0})} \xrightarrow{\sim} \mathrm{ind}_{KZ}^G\> V_0 \quad \text{ and } \quad \frac{\mathrm{ind}_{IZ}^G\> 1}{(1 + T_{1, 2}T_{1, 0})} \xrightarrow{\sim} \mathrm{ind}_{KZ}^G\> V_{p - 1}. \end{align} $$
These maps are respectively given by
Indeed, the first map sends 
to
$[\mathrm {id}, Y^{2p - 2} - X^{p - 1}Y^{p - 1}] \equiv [\mathrm {id}, X^{2p - 2} - X^{p - 1}Y^{p - 1} + Y^{2p - 2}]\ \mod \mathrm {ind}_{KZ}^G\> V_{p - 1}$
and the second map sends 
to
$[\mathrm {id}, Y^{2p - 2} - X^{p - 1}Y^{p - 1}] \equiv [\mathrm {id}, -X^{2p - 2}]\ \mod \mathrm {ind}_{KZ}^G\> V_0$
. Now using (6) and (3), we see that
$\mathrm {Im}\> T_{1, 2}T_{1, 0} \subseteq \mathrm {Ker}\> (T_{-1, 0} + T_{1, 0})$
and
$\mathrm {Im}\> (1 + T_{1, 2}T_{1, 0}) \subseteq \mathrm {Ker}\> T_{-1, 0}$
. Therefore, they are operators on the quotients
$$\begin{align*}\frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{1, 2}T_{1, 0})} \quad \text{ and } \quad \frac{\mathrm{ind}_{IZ}^G\> 1}{(1 + T_{1, 2}T_{1, 0})}, \end{align*}$$
respectively. Then, under the first map in (9), we see that
since
$\beta = \alpha w$
and
$w \in KZ$
. Similarly, under the second map
Remark 3.14
-
• In [Reference MorraMor13, Theorem
$1.4$
], Morra writes down the structure of the invariant vectors in
$\dfrac {\mathrm {ind}_{KZ}^G\> V_r}{(T)}$
for a certain principal congruence subgroup for
$0 \leq r \leq p - 1$
. Anandavardhanan and Borisagar later gave an alternate proof of this result for
$r \neq 0, p - 1$
, see [Reference Anandavardhanan and BorisagarAB13, Theorem
$1.2$
]. I thank the referee for pointing out the fact that Theorem 3.13 may be used to give an alternate proof of the
$r = 0, p - 1$
cases of Morra’s theorem. -
• It is well known that elements in
$\mathrm {ind}_{KZ}^G\> V_0$
are in bijection with functions on the vertices of
$\Delta $
with finite support. The element
$[g, 1]$
corresponds to the function that sends the vertex
$g \mathbb {Z}_p^2$
to
$1$
and all other vertices to
$0$
. In this spirit, one may ask what is the pictorial interpretation of the map In the language of functions on the oriented edges and vertices of the tree
$$ \begin{align*} \mathrm{ind}_{IZ}^G\> 1 \xrightarrow{} \mathrm{ind}_{KZ}^G\> V_0. \end{align*} $$
$\Delta $
, this map sends functions supported on edges to functions supported on the origins of those edges. For instance, if
$F$
is a function that maps an (oriented) edge
$e = (v_1, v_2)$
to
$1$
and all other (oriented) edges to
$0$
, then the map displayed above sends
$F$
to the function that maps
$v_1$
to
$1$
and all other vertices to
$0$
. Moreover, a short computation using the formulas of
$T_{1, 0}$
and
$T_{1, 2}$
in [Reference Barthel and LivnéBL95, Lemma 6] shows that on an edge
$e"$
, the value Since there are
$$\begin{align*}(T_{1, 2}T_{1, 0}F)(e") = \begin{cases} 1 & \text{ if } o(e") = o(e) \text{ and } e" \neq e, \\ 0 & \text{ otherwise.} \end{cases} \end{align*}$$
$p$
edges satisfying the conditions
$o(e") = o(e)$
and
$e" \neq e$
, we see that the isomorphism above sends
$T_{1, 2}T_{1, 0}F$
to
$0$
. Pictorially,
$F$
is supported on the red edge and
$T_{1, 2}T_{1, 0}F$
is supported on the blue edges in the following figure:Figure 1The tree for
$p = 2$
.
4 Proof of Theorem 2.2
In this section, we use Theorem 3.2 and Theorem 3.13 to rewrite Breuil’s mod
$p$
LLC as
Theorem 4.1 For
$r \in \{0, \ldots , p - 1\}$
,
$\lambda \in \overline {\mathbb {F}}_p$
and a smooth character
$\eta : \mathbb {Q}_p^* \to \overline {\mathbb {F}}_p^*$
, define
-
• If
$\lambda = 0:$
-
• If
$\lambda \neq 0:$
$$ \begin{align*}\!\! (\mu_{\lambda}\omega^{r + 1} \oplus \mu_{\lambda^{-1}})\otimes \eta \leftrightarrow & \left(\frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0} + \delta_{r, p - 1}T_{1, 0}) + (T_{1, 2} + \delta_{r, 0}T_{1, 0} - \lambda)} \otimes \eta \!\right)^{\!\!\mathrm{ss}} \oplus \\ & \left(\frac{\mathrm{ind}_{IZ}^G\> d^{[p - 3 - r]}}{(T_{-1, 0} + \delta_{[p - 3 - r], p - 1}T_{1, 0}) + (T_{1, 2} + \delta_{[p - 3 - r], 0}T_{1, 0} - \lambda^{-1})}\otimes \eta\omega^{r + 1}\!\right)^{\!\!\mathrm{ss}}. \end{align*} $$
Proof We only have to prove that for
$r \in \{0, 1, \ldots , p - 1\}$
,
$\lambda \in \overline {\mathbb {F}}_p$
and a smooth character
$\eta : \mathbb {Q}_p^* \to \overline {\mathbb {F}}_p^*$
,
$$\begin{align*}\pi(r, \lambda, \eta) \simeq \frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0} + \delta_{r, p - 1}T_{1, 0}) + (T_{1, 2} + \delta_{r, 0}T_{1, 0} - \lambda)} \otimes \eta. \end{align*}$$
First assume that
$r \in \{1, 2, \ldots , p - 2\}$
. Then using Theorem 3.2, there is an isomorphism
$$\begin{align*}\mathrm{ind}_{KZ}^G\> V_{r} \simeq \frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0})} \end{align*}$$
which sends the Hecke operator
$T$
to
$T_{1, 2}$
. Therefore, we see that
$$\begin{align*}\pi(r, \lambda, \eta) = \frac{\mathrm{ind}_{KZ}^G\> V_{r}}{T - \lambda}\otimes \eta \simeq \frac{\mathrm{ind}_{IZ}^G\> d^r}{(T_{-1, 0}) + (T_{1, 2} - \lambda)}\otimes \eta. \end{align*}$$
Next, assume that
$r = 0$
. Using Theorem 3.13, there is an isomorphism
$$\begin{align*}\mathrm{ind}_{KZ}^G\> V_0 \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{1, 2}T_{1, 0})} \end{align*}$$
under which the Hecke operator
$T$
corresponds to
$T_{-1, 0} + T_{1, 0}$
. Next, using Remark 3.11, we see that there is an isomorphism
$$\begin{align*}\frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{1, 2}T_{1, 0})} \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{-1, 0})} \end{align*}$$
under which
$T_{-1, 0} + T_{1, 0}$
on the left corresponds to
$T_{1, 2} + T_{1, 0}$
on the right. Therefore,
$$\begin{align*}\pi(0, \lambda, \eta) = \frac{\mathrm{ind}_{KZ}^G\> V_0}{(T - \lambda)}\otimes \eta \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{-1, 0}) + (T_{1, 2} + T_{1, 0} - \lambda)} \otimes \eta. \end{align*}$$
Finally, assume that
$r = p - 1$
. Using Theorem 3.13, there is an isomorphism
$$\begin{align*}\mathrm{ind}_{KZ}^G\> V_{p - 1} \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(1 + T_{1, 2}T_{1, 0})} \end{align*}$$
under which
$T$
corresponds to
$T_{-1, 0}$
. Again using Remark 3.11 as above, we have
$$\begin{align*}\frac{\mathrm{ind}_{IZ}^G\> 1}{(1 + T_{1, 2}T_{1, 0})} \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{-1, 0} + T_{1, 0})} \end{align*}$$
under which
$T_{-1, 0}$
on the left corresponds to
$T_{1, 2}$
on the right. Therefore,
$$\begin{align*}\pi(p - 1, \lambda, \eta) = \frac{\mathrm{ind}_{KZ}^G\> V_{p - 1}}{(T - \lambda)}\otimes \eta \simeq \frac{\mathrm{ind}_{IZ}^G\> 1}{(T_{-1, 0} + T_{1, 0}) + (T_{1, 2} - \lambda)}\otimes \eta.\\[-34pt] \end{align*}$$
Acknowledgments
I thank Prof. E. Ghate for assigning this problem to me and for many discussions. I also thank A. Jana for discussions on Iwahori–Hecke operators. I also thank the anonymous referee for their comments and suggestions.
Funding
During this work, I was partially supported by Harish-Chandra Research Institute.
