Notations

Throughout the paper, we fix an integer $n\geq 1$ and we write $\theta _{\mathrm {max}} := \lfloor \frac {n-1}{2} \rfloor $ so that $n = 2\theta _{\mathrm {max}}+1$ or $2(\theta _{\mathrm {max}}+1)$ according to whether n is odd or even. We also fix an odd prime number p. If k is a perfect field of characteristic p, we denote by $W(k)$ the ring of Witt vectors and by $W(k)_{{\mathbb {Q}}}$ its fraction field, which is an unramified extension of ${\mathbb {Q}}_p$ . We denote by $\sigma : x \mapsto x^p$ the Frobenius on k or its lift to $W(k)$ . If $q = p^e$ is a power of p, we write $\mathbb F_{q}$ for the field with q elements. In the special case where $q=p^2$ , we also use the alternative notation $\mathbb Z_{p^2} = W(\mathbb F_{p^2})$ and ${\mathbb {Q}}_{p^2} = W(\mathbb F_{p^2})_{{\mathbb {Q}}}$ . We fix an algebraic closure $\mathbb F$ of $\mathbb F_p$ . For $k\geq 1$ , the $k\times k$ identity matrix is denoted by $I_k$ , and the matrix with $1$ in the antidiagonal and $0$ everywhere else is denoted by $A_k$ . In various situations, the symbol $\mathbf 1$ will always represent the trivial representation of the group we are considering. The symmetric group of $\{1,\ldots ,k\}$ is denoted $\mathfrak S_k$ .

1 The Bruhat-Tits stratification on the PEL unitary Rapoport-Zink space of signature $(1,n-1)$

1.1 The PEL unitary Rapoport-Zink space $\mathcal M$ of signature $(1,n-1)$

In [Reference Vollaard and Wedhorn33], the authors introduce the PEL unitary Rapoport-Zink space $\mathcal M$ of signature $(1,n-1)$ as a moduli space, classifying the deformations of a given p-divisible group equipped with additional structures. We briefly recall the construction. Let E be a quadratic unramified extension of ${\mathbb {Q}}_p$ with ring of integers $\mathcal O_E$ and with nontrivial Galois involution $a \mapsto a^*$ . Let $\varphi _0: K \xrightarrow {\sim } {\mathbb {Q}}_{p^2}$ be a ${\mathbb {Q}}_p$ -linear isomorphism and let $\varphi _1 := \sigma \circ \varphi _0$ . Let $\mathrm {Nilp}$ denote the category of schemes over $\mathbb Z_{p^2}$ where p is locally nilpotent. For $S\in \mathrm {Nilp}$ , a unitary p -divisible group of signature $(1,n-1)$ over S is a triple $(X,\iota _X,\lambda _X)$ where

1. – X is a p-divisible group over S.

2. – $\iota _X: \mathcal O_E\rightarrow \mathrm {End}(X)$ is a $\mathcal O_E$ -action on X such that the induced action on its Lie algebra satisfies the signature $(1,n-1)$ condition: for every $a \in \mathcal O_E$ , the characteristic polynomial of $\iota _X(a)$ acting on $\mathrm {Lie}(X)$ is given by

   $$ \begin{align*}(T-\varphi_0(a))^1(T-\varphi_1(a))^{n-1} \in \mathbb Z_{p^2}[T] \subset \mathcal O_{S}[T].\end{align*} $$

3. – $\lambda _X:X \xrightarrow {\sim } {}^tX$ is an $\mathcal O_E$ -linear polarization where ${}^tX$ denotes the Serre dual of X.

The $\mathcal O_E$ -linearity of $\lambda _X$ is with respect to the $\mathcal O_E$ -actions $\iota _X$ and the induced action $\iota _{{}^tX}$ on the dual. A specific example of unitary p-divisible group over $\mathbb F_{p^2}$ is given in [Reference Vollaard and Wedhorn33] 2.4 by means of covariant Dieudonné theory. We denote it by $(\mathbb X,\iota _{\mathbb X},\lambda _{\mathbb X})$ and call it the standard unitary p -divisible group. The p-divisible group $\mathbb X$ is superspecial. The following set-valued functor $\mathcal M$ defines a moduli problem classifying deformations of $\mathbb X$ by quasi-isogenies. More precisely, for $S \in \mathrm {Nilp}$ the set $\mathcal M(S)$ consists of all isomorphism classes of tuples $(X,\iota _X,\lambda _X,\rho _X)$ such that

1. – $(X,\lambda _X,\rho _X)$ is a unitary p-divisible group of signature $(1,n-1)$ over S.

2. – $\rho _X: X\times _S \overline {S} \rightarrow \mathbb X\times _{\mathbb F_{p^2}} \overline {S}$ is an $\mathcal O_{E}$ -linear quasi-isogeny compatible with the polarizations, in the sense that ${}^t\rho _X \circ \lambda _{\mathbb X} \circ \rho _X$ is a ${\mathbb {Q}}_p^{\times }$ -multiple of $\lambda _X$ .

In the second condition, $\overline {S}$ denotes the special fiber of S. By [Reference Rapoport and Zink28] Corollary 3.40, this moduli problem is represented by a separated formal scheme $\mathcal M$ over $\mathrm {Spf}(\mathbb Z_{p^2})$ , called a Rapoport-Zink space. It is formally locally of finite type, and since the associated PEL datum is unramified it is also formally smooth over $\mathbb Z_{p^2}$ . The reduced special fiber of $\mathcal M$ is the reduced $\mathbb F_{p^2}$ -scheme ${\mathcal {M}}_{\mathrm {red}}$ defined by the maximal ideal of definition. Rational points of $\mathcal M$ over a perfect field extension k of $\mathbb F_{p^2}$ can be understood in terms of semi-linear algebra by means of Dieudonné theory. We denote by $M(\mathbb X)$ the (covariant) Dieudonné module of $\mathbb X$ , this is a free $\mathbb Z_{p^2}$ -module of rank $2n$ . We denote by $N(\mathbb X) := M(\mathbb X)\otimes {\mathbb {Q}}_{p^2}$ its isocrystal. By construction, the Frobenius $\mathbf F$ and the Verschiebung $\mathbf V$ agree on $N(\mathbb X)$ . In particular, we have $\mathbf F^2 = p\cdot \mathrm {id}$ on the isocrystal. The $\mathcal O_E$ -action $\iota _{\mathbb X}$ induces a $\mathbb Z/2\mathbb Z$ -grading $M(\mathbb X) = M(\mathbb X)_0 \oplus M(\mathbb X)_1$ as a sum of two free $\mathbb Z_{p^2}$ -modules of rank n, such that $a \in \mathcal O_E$ acts via $\varphi _i(a)$ on $M(\mathbb X)_i$ for $i = 0,1$ . The same goes for the isocrystal $N(\mathbb X) = N(\mathbb X)_0 \oplus N(\mathbb X)_1$ where $N(\mathbb X)_i = M(\mathbb X)_i\otimes {\mathbb {Q}}_{p^2}$ for $i=0,1$ . The polarization $\lambda _{\mathbb X}$ induces a perfect alternating ${\mathbb {Q}}_{p^2}$ -bilinear pairing $\langle \cdot ,\cdot \rangle $ on $N(\mathbb X)$ such that

$$ \begin{align*} \forall x,y \in N(\mathbb X), \forall a \in E, & & \langle \mathbf Fx,y\rangle = \langle x,\mathbf Vy\rangle^{\sigma} \text{ and } \langle ax,y \rangle = \langle x, a^*y\rangle. \end{align*} $$

Moreover $\langle \cdot ,\cdot \rangle $ restricts to a perfect $\mathbb Z_{p^2}$ -pairing on the lattice $M(\mathbb X)$ . The pieces $N(\mathbb X)_i$ are totally isotropic for $i=0,1$ and dual of each other. Moreover, the Frobenius $\mathbf F$ is $1$ -homogeneous with respect to this grading. As in [Reference Vollaard and Wedhorn33] 2.6, we define

$$ \begin{align*} \forall x,y \in N(\mathbb X)_0, && \{x,y\} := \delta\langle x,\mathbf Fy\rangle, \end{align*} $$

where $\delta \in \mathbb Z_{p^2}^{\times }$ is a scalar satisfying $\delta ^{\sigma } = -\delta $ . The pairing $\{\cdot ,\cdot \}$ is a perfect $\sigma $ -hermitian form on $N(\mathbb X)_0$ .

Notation. From now on, we will write $\mathbf V := N(\mathbb X)_0$ and $\mathbf M := M(\mathbb X)_0$ .

Then $\mathbf V$ is a ${\mathbb {Q}}_{p^2}$ -hermitian space of dimension n, and $\mathbf M$ is a given $\mathbb Z_{p^2}$ -lattice, i.e. a finitely generated $\mathbb Z_{p^2}$ -submodule containing a basis of $\mathbf V$ . Given two lattices $M_1$ and $M_2$ , the notation $M_1 \overset {d}{\subset } M_2$ means that $M_1\subset M_2$ and the quotient module $M_2/M_1$ has length d. The integer d is called the index of $M_1$ in $M_2$ , and is denoted $d = [M_2:M_1]$ . Given a lattice $M\subset \mathbf V$ , we define the dual lattice $M^{\vee } := \{v \in \mathbf V \,|\, \{v,M\} \subset \mathbb Z_{p^2}\}$ .By construction the lattice $\mathbf M$ satisfies

$$ \begin{align*}p\mathbf M^{\vee} \overset{1}{\subset} \mathbf M \overset{n-1}{\subset} \mathbf M^{\vee}.\end{align*} $$

Consider the matrices

$$ \begin{align*}T_{\text{odd}}:= A_{2\theta_{\mathrm{max}}+1}, \quad \quad T_{\text{even}}:= \left(\begin{matrix} & & & A_{\theta_{\mathrm{max}}} \\ & 1 & 0 & \\ & 0 & p & \\ A_{\theta_{\mathrm{max}}} & & & \end{matrix} \right). \end{align*} $$

By [Reference Vollaard32] Proposition 1.15, there exists a basis of $\mathbf V$ such that $\{\cdot ,\cdot \}$ is represented by the matrix $T_{\text {odd}}$ if n is odd and by $T_{\text {even}}$ if n is even. A Witt decomposition on $\mathbf V$ is a set $\{L_i\}_{i\in I}$ of isotropic lines in $\mathbf V$ such that the following conditions are satisfied:

1. – for every $i \in I$ , there is a unique $i'\in I$ such that $\{L_i,L_{i'}\} \not = 0$ ,

2. – the sum of the $L_i$ ’s is direct,

3. – the orthogonal of the direct sum of the $L_i$ ’s is an anisotropic subspace of $\mathbf V$ .

Since each line $L_i$ is isotropic, in the first condition one necessarily has $(i')' = i$ and $i \not = i'$ . As a consequence, we have $\#I = 2w(\mathbf V)$ for some integer $w = w(\mathbf V)$ called the Witt index of $\mathbf V$ . It does not depend on the choice of a Witt decomposition. We write $L^{\mathrm {an}}$ for the orthogonal of the direct sum of the $L_i$ ’s. The dimension of $L^{\mathrm {an}}$ is $n^{\mathrm {an}} := n - 2w$ . Given a Witt decomposition of $\mathbf V$ , one may find vectors $e_i \in L_i$ such that $\{e_i,e_j\} = \delta _{j,i'}$ . Together with a choice of an orthogonal basis for $L^{\mathrm {an}}$ , these vectors define a basis of $\mathbf V$ which is said to be adapted to the Witt decomposition. For any $i\in I$ , the direct sum $L_i \oplus L_{i'}$ is isometric to the hyperbolic plane $\mathbf H$ . Therefore, we obtain a decomposition

$$ \begin{align*}\mathbf V = w\mathbf H \oplus L^{\mathrm{an}}.\end{align*} $$

We may always rearrange the index set so that $I = \{-w,\ldots ,-1,1,\ldots ,w\}$ and $i' = -i$ for all $i \in I$ . In this context, we write $L_0$ instead of $L^{\mathrm {an}}$ .

We fix once and for all a basis e of $\mathbf V$ in which the hermitian form is represented by the matrix $T_{\text {odd}}$ or $T_{\text {even}}$ . In the case $n = 2\theta _{\mathrm {max}}+1$ is odd, we will denote it

$$ \begin{align*}e = (e_{-\theta_{\mathrm{max}}} , \ldots , e_{-1} , e_0^{\mathrm{an}} , e_1 , \ldots , e_{\theta_{\mathrm{max}}}),\end{align*} $$

and in the case $n = 2(\theta _{\mathrm {max}}+1)$ is even we will denote it

$$ \begin{align*}e = (e_{-\theta_{\mathrm{max}}} , \ldots , e_{-1} , e_0^{\mathrm{an}} , e_1^{\mathrm{an}} , e_1 , \ldots , e_{\theta_{\mathrm{max}}}).\end{align*} $$

The choice of such a basis gives a Witt decomposition with $L_i := {\mathbb {Q}}_{p^2}e_i$ and $L_0$ the subspace generated by $e_0^{\mathrm {an}}$ , and when n is even by $e_1^{\mathrm {an}}$ as well. In particular, $w(\mathbf V) = \theta _{\mathrm {max}}$ and $n^{\mathrm {an}} = 1$ or $2$ depending on whether n is odd or even, respectively.

Given a perfect field extension k of $\mathbb F_{p^2}$ , we denote by $\mathbf V_k$ the base change $\mathbf V \otimes _{{\mathbb {Q}}_{p^2}} W(k)_{{\mathbb {Q}}}$ . The form may be extended to $\mathbf V_k$ by the formula

$$ \begin{align*}\{v\otimes x,w\otimes y\} := xy^{\sigma}\{v,w\}\in W(k)_{{\mathbb{Q}}}\end{align*} $$

for all $v,w\in \mathbf V$ and $x,y\in W(k)_{{\mathbb {Q}}}$ . The notions of index and duality for $W(k)$ -lattices can be extended as well. By [Reference Vollaard32] Proposition 1.10, the rational points of the Rapoport-Zink space are described by the following statement.

Proposition 1.1. Let k be a perfect field extension of $\mathbb F_{p^2}$ . There is a natural bijection between $\mathcal M(k)$ and the set of $W(k)$ -lattices M in $\mathbf V_k$ such that for some integer $i\in \mathbb Z$ , we have

$$ \begin{align*}p^{i+1}M^{\vee} \overset{1}{\subset} M \overset{n-1}{\subset} p^i M^{\vee}. \end{align*} $$

There is a decomposition $\mathcal M = \bigsqcup _{i\in \mathbb Z} {\mathcal {M}}_i$ into formal connected subschemes which are open and closed. The rational points of ${\mathcal {M}}_i$ are those lattices M satisfying the relation above with the given integer i. In particular, the lattice $\mathbf M$ defined in the previous paragraph is an element of ${\mathcal {M}}_0(\mathbb F_{p^2})$ . By [Reference Vollaard32] Proposition 1.7, the formal scheme ${\mathcal {M}}_i$ is empty if $ni$ is odd.

Let $J = \mathrm {GU}(\mathbf V)$ be the group of unitary similitudes of $\mathbf V$ , seen as a reductive group over ${\mathbb {Q}}_p$ . Then $J({\mathbb {Q}}_p)$ consists of all $g\in \mathrm {GL}_{{\mathbb {Q}}_{p^2}}(\mathbf V)$ which preserve the hermitian form up to a unit $c(g)\in {\mathbb {Q}}_{p}^{\times }$ , called the multiplier. By Dieudonné theory, the group $J({\mathbb {Q}}_p)$ is also identified with the group of quasi-isogenies $\mathbb X \to \mathbb X$ of unitary p-divisible groups. The space $\mathcal M$ is endowed with a natural action of $J({\mathbb {Q}}_p)$ . At the level of points, the element g acts by sending a lattice M to $g(M)$ . For $g\in J({\mathbb {Q}}_p)$ , let $\alpha (g)$ be the p-adic valuation of $c(g)$ . This defines a continuous homomorphism

$$ \begin{align*}\alpha: J\rightarrow \mathbb Z\end{align*} $$

where $\mathbb Z$ is given the discrete topology. Then g induces an isomorphism ${\mathcal {M}}_i \xrightarrow {\sim } {\mathcal {M}}_{i+\alpha (g)}$ . According to [Reference Vollaard32] 1.17 the image of $\alpha $ is $\mathbb Z$ if n is even, and $2\mathbb Z$ if n is odd. The centre $\mathrm Z(J({\mathbb {Q}}_p))$ consists of all the scalar matrices, so that it is identified with ${\mathbb {Q}}_{p^2}^{\times }$ . If $\lambda \in {\mathbb {Q}}_{p^2}^{\times }$ , then $c(\lambda \cdot \mathrm {id}) = \mathrm {Norm}(\lambda ) \in {\mathbb {Q}}_{p}^{\times }$ , where $\mathrm {Norm}$ is the norm map relative to the quadratic extension ${\mathbb {Q}}_{p^2}/{\mathbb {Q}}_{p}$ . In particular, $\alpha (\mathrm Z(J)) = 2\mathbb Z$ . Thus, the restriction of $\alpha $ to the centre is surjective onto $\mathrm {Im}(\alpha )$ only when n is odd. When n is even, we define the following element

$$ \begin{align*}g_0 := \left(\begin{matrix} & & & I_{\theta_{\mathrm{max}}} \\ & 0 & p & \\ & 1 & 0 & \\ pI_{\theta_{\mathrm{max}}} & & & \end{matrix} \right).\end{align*} $$

Then $g_0 \in J({\mathbb {Q}}_p)$ and $c(g_0) = p$ so that $\alpha (g_0) = 1$ . Moreover $g_0^2 = p\cdot \mathrm {id}$ belongs to $\mathrm Z(J({\mathbb {Q}}_p))$ . Let i and $i'$ be two integers such that $ni$ and $ni'$ are even. We consider the multiplication $p^{\frac {i'-i}{2}}:\mathbb X\to \mathbb X$ when $i\equiv i' \mod 2$ , and the quasi-isogeny $p^{\frac {i'-i-1}{2}}g_0:\mathbb X \to \mathbb X$ when $i\not \equiv i' \mod 2$ . This is well defined as the second case may only happen when n is even. It induces a morphism $\psi _{i,i'}:{\mathcal {M}}_i \rightarrow {\mathcal {M}}_{i'}$ . By [Reference Vollaard32] Proposition 1.18, the map $\psi _{i,i'}$ is an isomorphism between ${\mathcal {M}}_i$ and ${\mathcal {M}}_{i'}$ , and if $i,i'$ and $i"$ are three integers such that $ni, ni'$ and $ni"$ are even, then we have $\psi _{i',i"}\circ \psi _{i,i'} = \psi _{i,i"}$ .

1.2 The Bruhat-Tits stratification of the special fiber ${\mathcal {M}}_{\mathrm {red}}$

We now recall the construction of the Bruhat-Tits stratification on ${\mathcal {M}}_{\mathrm {red}}$ as in [Reference Vollaard and Wedhorn33]. Let i be an integer such that $ni$ is even. We define

$$ \begin{align*}{\mathcal{L}}_i := \{\Lambda\subset \mathbf V \text{ a } \mathbb Z_{p^2}-\text{lattice}\,|\, p^{i+1}\Lambda^{\vee}\subsetneq \Lambda \subset p^i\Lambda^{\vee}\}.\end{align*} $$

If $\Lambda \in {\mathcal {L}}_i$ , we define its orbit type $t(\Lambda ):= [\Lambda :p^{i+1}\Lambda ^{\vee }]$ . We also call it the type of $\Lambda $ . In particular, the lattices in ${\mathcal {L}}_i$ of type $1$ are precisely the $\mathbb F_{p^2}$ -rational points of ${\mathcal {M}}_{i}$ . By sending $\Lambda $ to $g(\Lambda )$ , an element $g\in J({\mathbb {Q}}_p)$ defines a map ${\mathcal {L}}_i\rightarrow {\mathcal {L}}_{i+\alpha (g)}$ . The following Proposition follows from [Reference Vollaard32] Remark 2.3 and [Reference Vollaard and Wedhorn33] Remark 4.1.

Proposition 1.2. Let i be an integer such that $ni$ is even and let $\Lambda \in {\mathcal {L}}_i$ .

1. – The map ${\mathcal {L}}_i\rightarrow {\mathcal {L}}_{i+\alpha (g)}$ induced by an element $g\in J({\mathbb {Q}}_p)$ is an inclusion preserving, type preserving bijection.

2. – We have $1\leq t(\Lambda ) \leq n$ . Furthermore $t(\Lambda )$ is odd.

3. – The sets ${\mathcal {L}}_i$ ’s for various i’s are pairwise disjoint.

Moreover, two lattices $\Lambda , \Lambda ' \in \bigsqcup _{ni \in 2\mathbb Z}{\mathcal {L}}_i$ are in the same orbit under the action of $J({\mathbb {Q}}_p)$ if and only if $t(\Lambda ) = t(\Lambda ')$ .

We write $\mathcal L := \bigsqcup _{ni \in 2\mathbb Z}{\mathcal {L}}_i$ . For any odd number t between $1$ and n, there exists a lattice $\Lambda \in {\mathcal {L}}_0$ of orbit type t. Write $t_{\mathrm {max}} := 2\theta _{\mathrm {max}}+1$ , so that the orbit type t of any lattice in $\mathcal L$ satisfies $1\leq t \leq t_{\mathrm {max}}$ . The following lemma will be useful later.

Lemma 1.3. Let $i \in \mathbb Z$ such that $ni$ is even, and let $\Lambda \in {\mathcal {L}}_i$ . We have $\Lambda ^{\vee } \in \mathcal L$ if and only if either n is even or n is odd and $t(\Lambda ) = t_{\mathrm {max}}$ . If $\Lambda ^{\vee } \in \mathcal L$ and n is even, then $\Lambda ^{\vee } \in {\mathcal {L}}_{-i-1}$ and $t(\Lambda ^{\vee }) = n - t(\Lambda )$ . If on the contrary n is odd, then $\Lambda ^{\vee } \in {\mathcal {L}}_{-i}$ and $t(\Lambda ^{\vee }) = t(\Lambda )$ .

Proof. First we prove the converse. We have the following chain of inclusions

$$ \begin{align*}p^{-i}\Lambda \overset{n-t(\Lambda)}{\subset} \Lambda^{\vee} \overset{t(\Lambda)}{\subset} p^{-i-1}\Lambda.\end{align*} $$

If n is even, then $-n(i+1)$ is also even and $n-t(\Lambda ) \not = 0$ . Since $(\Lambda ^{\vee })^{\vee } = \Lambda $ , we deduce that $\Lambda ^{\vee } \in {\mathcal {L}}_{-i-1}$ with orbit type $n-t(\Lambda )$ . Assume now that n is odd and that $t(\Lambda ) = t_{\mathrm {max}} = n$ . Then $\Lambda ^{\vee } = p^{-i}\Lambda \in {\mathcal {L}}_{-i}$ .

Let us now assume that $\Lambda ^{\vee } \in \mathcal L$ and that n is odd. Let $i' \in 2\mathbb Z$ such that $\Lambda ^{\vee } \in {\mathcal {L}}_{i'}$ . We have

$$ \begin{align*} \Lambda^{\vee} \overset{n-t(\Lambda^{\vee})}{\subset} p^{i'}\Lambda \overset{n-t(\Lambda)}{\subset} p^{i'+i}\Lambda^{\vee}, & & \Lambda^{\vee} \overset{t(\Lambda)}{\subset} p^{-i-1}\Lambda \overset{t(\Lambda^{\vee})}{\subset} p^{-i-i'-2}\Lambda^{\vee}, \end{align*} $$

therefore $-2\leq i+i' \leq 0$ . Since $i+i'$ is even it is either $-2$ or $0$ . If it were $-2$ , then we would have $t(\Lambda ) = t(\Lambda ^{\vee }) = 0$ which is absurd. Therefore $i+i' = 0$ , and we have $n-t(\Lambda ) = n - t(\Lambda ^{\vee }) = 0$ .

With the help of ${\mathcal {L}}_i$ , one may construct an abstract simplicial complex $\mathcal B_i$ . For $s\geq 0$ , an s-simplex of $\mathcal B_i$ is a subset $S\subset {\mathcal {L}}_i$ of cardinality $s+1$ such that for some ordering $\Lambda _0,\ldots ,\Lambda _s$ of its elements, we have a chain of inclusions $p^{i+1}\Lambda _s^{\vee }\subsetneq \Lambda _0 \subsetneq \Lambda _1 \subsetneq \ldots \subsetneq \Lambda _s$ . We must have $0\leq s \leq m$ for such a simplex to exist. Let $\tilde {J} = \mathrm {SU}(\mathbf V)$ be the derived group of J. We consider the abstract simplicial complex $\mathrm {BT}(\tilde J,{\mathbb {Q}}_p)$ of the Bruhat-Tits building of $\tilde {J}$ over ${\mathbb {Q}}_p$ . A concrete description of this complex is given in [Reference Vollaard32] Theorem 3.5.

Theorem 1.4. The Bruhat-Tits building $\mathrm {BT}(\tilde J,{\mathbb {Q}}_p)$ is naturally identified with $\mathcal B_i$ for any fixed integer i such that $ni$ is even. The set ${\mathcal {L}}_i$ is identified with the set of vertices of $\mathrm {BT}(\tilde J,{\mathbb {Q}}_p)$ . The identification is $\tilde J({\mathbb {Q}}_p)$ -equivariant.

Apartments in the Bruhat-Tits building $\mathrm {BT}(\tilde J,{\mathbb {Q}}_p)$ are in one-to-one correspondence with Witt decompositions of $\mathbf V$ . Let $L = \{L_i\}_{i\in I}$ be a Witt decomposition of $\mathbf V$ and let $f = (f_i)_{i\in I} \sqcup B^{\mathrm {an}}$ be a basis of $\mathbf V$ adapted to the decomposition, where $f_i \in L_i$ and $B^{\mathrm {an}}$ is an orthogonal basis of $L^{\mathrm {an}}$ . Under the identification of $\mathrm {BT}(\tilde J,{\mathbb {Q}}_p)$ with $\mathcal B_i$ , the vertices inside the apartment associated to L correspond to the lattices $\Lambda \in {\mathcal {L}}_i$ which are equal to the direct sum of $\Lambda \cap L^{\mathrm {an}}$ and of the modules $p^{r_i}\mathbb Z_{p^2}f_i$ for some integers $(r_i)_{i\in I}$ . The subset of ${\mathcal {L}}_i$ consisting of all such lattices will be denoted ${\mathcal {A}}_i^L$ or, with an abuse of notations, ${\mathcal {A}}_i^f$ . We call such a set ${\mathcal {A}}_i^L$ the apartment associated to L in ${\mathcal {L}}_i$ . We also define $\mathcal A^L := \bigsqcup _{ni\in 2\mathbb Z} {\mathcal {A}}_i^L$ . We recall a general result regarding Bruhat-Tits buildings.

Proposition 1.5. Let i be an integer such that $ni$ is even. Any two lattices $\Lambda $ and $\Lambda '$ in ${\mathcal {L}}_i$ lie inside a common apartment ${\mathcal {A}}_i^{L}$ for some Witt decomposition L. Moreover, the action of $\tilde J({\mathbb {Q}}_p)$ acts transitively on the set of apartments $\{{\mathcal {A}}_i^L\}_L$ .

Recall the basis e of $\mathbf V$ that we have fixed earlier. We will denote by

$$ \begin{align*}\Lambda(r_{-\theta_{\mathrm{max}}},\ldots ,r_{-1},s,r_1,\ldots ,r_{\theta_{\mathrm{max}}})\end{align*} $$

the $\mathbb Z_{p^2}$ -lattice generated by the vectors $p^{r_j}e_j$ for all $j = \pm 1,\ldots , \pm \theta _{\mathrm {max}}$ , by $p^{s_0}e^{\mathrm {an}}_0$ and if n is even, by $p^{s_1}e^{\mathrm {an}}_1$ too. Here, the $r_j$ ’s are integers and s denotes either the integer $s_0$ if n is odd or the pair of integers $(s_0,s_1)$ if n is even.

Proposition 1.6. Let i be an integer such that $ni$ is even. Let $(r_j,s)$ be a family of integers as above. The corresponding lattice $\Lambda = \Lambda (r_{-\theta _{\mathrm {max}}},\ldots ,r_{-1},s,r_1,\ldots ,r_{\theta _{\mathrm {max}}})$ belongs to ${\mathcal {L}}_i$ if and only if the following conditions are satisfied

1. – for all $1\leq j \leq \theta _{\mathrm {max}}$ , we have $r_{-j} + r_j \in \{i,i+1\}$ ,

2. – $s_0 = \lfloor \frac {i+1}{2}\rfloor $ ,

3. – if n is even, then $s_1 = \lfloor \frac {i}{2}\rfloor $ .

Moreover, when $\Lambda \in {\mathcal {L}}_i$ then

$$ \begin{align*}t(\Lambda) = 1 + 2\#\{1\leq j \leq \theta_{\mathrm{max}} \,|\, r_{-j} + r_j = i\}.\end{align*} $$

Proof. The lattice $\Lambda $ belongs to ${\mathcal {L}}_i$ if and only if the following chain of inclusions holds

$$ \begin{align*}p^{i+1}\Lambda^{\vee} \subsetneq \Lambda \subset p^i\Lambda^{\vee}.\end{align*} $$

The dual lattice $\Lambda ^{\vee }$ is equal to the lattice $\Lambda (-r_{\theta _{\mathrm {max}}},\ldots ,-r_{1},s',-r_{-1},\ldots ,-r_{-\theta _{\mathrm {max}}})$ , where $s' = -s_0$ when n is odd, and $s' = (-s_0,-s_1-1)$ when n is even. Thus, the inclusions above are equivalent to the following inequalities:

$$ \begin{align*} & i - r_{-j} \leq r_j \leq i + 1 - r_{-j}, & i - s_0 \leq s_0 \leq i + 1 - s_0,\\ & i - 1 - s_1 \leq s_1 \leq i - s_1 \text{ (if }n\text{ is even)}. & \end{align*} $$

This proves the desired condition on the integers $r_j$ ’s and on s. Let us now assume that $\Lambda \in {\mathcal {L}}_i$ . Its orbit type is equal to the index $[\Lambda ,p^{i+1}\Lambda ^{\vee }]$ . This corresponds to the number of times equality occurs with the left-hand side in all the inequalities above. Of course, if the equality $i - r_{-j} = r_j$ occurs for some j, then it occurs also for $-j$ . Moreover, if i is even then the equality $i - s_0 = s_0$ occurs whereas $i - 1 - s_1 \not = s_1$ . On the contrary if i is odd, then the equality $i-1-s_1 = s_1$ occurs whereas $i - s_0 \not = s_0$ . Thus in all cases, only one of $s_0$ and $s_1$ contributes to the index. Putting things together, we deduce the desired formula.

We deduce the following corollary.

Corollary 1.7. The apartment $A_i^e$ (resp. $A^e$ ) consists of all the lattices of the form

$$ \begin{align*}\Lambda = \Lambda(r_{-\theta_{\mathrm{max}}},\ldots ,r_{-1},s,r_1,\ldots ,r_{\theta_{\mathrm{max}}})\end{align*} $$

which belong to ${\mathcal {L}}_i$ (resp. to $\mathcal L$ ).

Proof. According to the previous proposition, it is clear that all lattices which belong to ${\mathcal {L}}_i$ and are of the form $\Lambda (r_{-\theta _{\mathrm {max}}},\ldots ,r_{-1},s,r_1,\ldots ,r_{\theta _{\mathrm {max}}})$ are elements of ${\mathcal {A}}_i^e$ . We shall prove the converse. Let $\Lambda \in {\mathcal {A}}_i^e$ . By definition, there exists integers $(r_j)_j$ such that

$$ \begin{align*}\Lambda = \Lambda\cap\mathbf V^{\mathrm{an}} \oplus \bigoplus_{1\leq j \leq \theta_{\mathrm{max}}}\left(p^{r_{-j}}\mathbb Z_{p^2}e_{-j} \oplus p^{r_j}\mathbb Z_{p^2}e_j\right).\end{align*} $$

Write $\Lambda ' = \Lambda \cap \mathbf V^{\mathrm {an}}$ . This is a lattice in $\mathbf V^{\mathrm {an}}$ which satisfies the chain of inclusions

$$ \begin{align*}p^{i+1}\Lambda'\,^{\vee} \subset \Lambda' \subset p^i\Lambda'\,^{\vee},\end{align*} $$

where the duals are taken with respect to the restriction of $\{\cdot ,\cdot \}$ to $\mathbf V^{\mathrm {an}}$ . Since $\mathbf V^{\mathrm {an}}$ is anisotropic, there is only a single lattice satisfying the chain of inclusions above. If we write $a := \lfloor \frac {i+1}{2}\rfloor $ and $b := \lfloor \frac {i}{2}\rfloor $ , it is given by $p^{a}\mathbb Z_{p^2}e_0^{\mathrm {an}}$ if n is odd, and by $p^{a}\mathbb Z_{p^2}e_0^{\mathrm {an}} \oplus p^{b}\mathbb Z_{p^2}e_1^{\mathrm {an}}$ if n is even. Thus, it must be equal to $\Lambda '$ and it concludes the proof.

We fix a maximal simplex in ${\mathcal {L}}_0$ lying inside the apartment ${\mathcal {A}}_0^e$ . For $0\leq \theta \leq \theta _{\mathrm {max}}$ we define

$$ \begin{align*}\Lambda_{\theta} := \Lambda(\underbrace{0,\ldots,0}_{\theta_{\mathrm{max}}},0,\underbrace{0,\ldots,0}_{\theta}, \underbrace{1,\ldots ,1}_{\theta_{\mathrm{max}}-\theta}).\end{align*} $$

Here, the $0$ in the middle stands for $(0,0)$ in case n is even. We have $t(\Lambda _{\theta }) = 2\theta +1$ and

$$ \begin{align*}p\Lambda_0^{\vee}\subsetneq \Lambda_0 \subset \ldots \subset \Lambda_{\theta_{\mathrm{max}}}.\end{align*} $$

Thus, they form an $\theta _{\mathrm {max}}$ -simplex in ${\mathcal {L}}_0$ . Given a lattice $\Lambda \in {\mathcal {L}}_i$ , a subfunctor ${\mathcal {M}}_{\Lambda }$ of ${\mathcal {M}}_{i,\mathrm {red}}$ is defined in [Reference Vollaard and Wedhorn33], classifying those p-divisible groups for which a certain quasi-isogeny, depending on $\Lambda $ , is in fact an actual isogeny. In Lemma 4.2, the authors prove that it is representable by a projective scheme over $\mathbb F_{p^2}$ , and that the natural morphism ${\mathcal {M}}_{\Lambda } \hookrightarrow {\mathcal {M}}_{i,\mathrm {red}}$ is a closed immersion. The schemes ${\mathcal {M}}_{\Lambda }$ are called the closed Bruhat-Tits strata of $\mathcal M$ . Their rational points are described as follows, see Lemma 4.3 of loc. cit.

Proposition 1.8. Let k be a perfect field extension of $\mathbb F_{p^2}$ , and let $M\in {\mathcal {M}}_{i,\mathrm {red}}(k)$ . Then

$$ \begin{align*}M\in {\mathcal{M}}_{\Lambda}(k) \iff M \subset \Lambda_k := \Lambda \otimes_{\mathbb Z_{p^2}} W(k).\end{align*} $$

The set of lattices satisfying the condition above was conjectured in [Reference Vollaard32] to be the set of points of a subscheme of ${\mathcal {M}}_{i,\mathrm {red}}$ , and it was proved in the special cases $n=2,3$ . In [Reference Vollaard and Wedhorn33], the general argument is given by the construction of ${\mathcal {M}}_{\Lambda }$ . The action of an element $g\in J({\mathbb {Q}}_p)$ on $\mathcal {M}_{\mathrm {red}}$ induces an isomorphism ${\mathcal {M}}_{\Lambda } \xrightarrow {\sim } {\mathcal {M}}_{g\cdot \Lambda }$ .

Let $\Lambda \in \mathcal L$ . We denote by $J_{\Lambda }$ the fixator of $\Lambda $ under the action of $J({\mathbb {Q}}_p)$ . If $\Lambda = \Lambda _{\theta }$ for some $0\leq \theta \leq \theta _{\mathrm {max}}$ , we will write $J_{\theta }$ instead. These are maximal parahoric subgroups of $J({\mathbb {Q}}_p)$ . In unramified unitary similitude groups, maximal parahoric subgroups and maximal compact subgroups are the same. A general parahoric subgroup is an intersection $J_{\Lambda _1}\cap \ldots \cap J_{\Lambda _s}$ where $\{\Lambda _1,\ldots ,\Lambda _s\}$ is an s-simplex in ${\mathcal {L}}_i$ for some i. Any parahoric subgroup is compact and open in $J({\mathbb {Q}}_p)$ . Let i be the integer such that $\Lambda \in {\mathcal {L}}_i$ . We define $V_{\Lambda }^0:= \Lambda /p^{i+1}\Lambda ^{\vee }$ and $V_{\Lambda }^1 := p^{i}\Lambda ^{\vee }/\Lambda $ . Since $p\Lambda \subset p\cdot p^i\Lambda ^{\vee }$ and $p\cdot p^{i}\Lambda ^{\vee } \subset \Lambda $ , these are both $\mathbb F_{p^2}$ -vector space of dimensions, respectively, $t(\Lambda )$ and $n-t(\Lambda )$ . Both spaces come together with a non-degenerate $\sigma $ -hermitian form $(\cdot ,\cdot )_0$ and $(\cdot ,\cdot )_1$ with values in $\mathbb F_{p^2}$ , respectively induced by $p^{-i}\{\cdot ,\cdot \}$ and by $p^{-i+1}\{\cdot ,\cdot \}$ . If k is a perfect field extension of $\mathbb F_{p^2}$ and if $\epsilon \in \{0,1\}$ , we may extend the pairings to $(V_{\Lambda }^{\epsilon })_k = V_{\Lambda }^{\epsilon }\otimes _{\mathbb F_{p^2}} k$ by setting

$$ \begin{align*}(v\otimes x,w\otimes y)_{\epsilon}:=xy^{\sigma}(v,w)_{\epsilon}\in k\end{align*} $$

for all $v,w\in V_{\Lambda }^{\epsilon }$ and $x,y\in k$ . If U is a subspace of $(V_{\Lambda }^{\epsilon })_k$ we denote by $U^{\perp }$ its orthogonal.

Denote by $J_{\Lambda }^+$ the pro-unipotent radical of $J_{\Lambda }$ and write ${\mathcal {J}}_{\Lambda } := J_{\Lambda }/J_{\Lambda }^+$ . This is a finite group of Lie type, called the maximal reductive quotient of $J_{\Lambda }$ . We have an identification ${\mathcal {J}}_{\Lambda } \simeq \mathrm {G}(\mathrm {U}(V_{\Lambda }^0) \times \mathrm {U}(V_{\Lambda }^1))$ , that is the group of pairs $(g_0,g_1)$ where for $\epsilon \in \{0,1\}$ we have $g_{\epsilon } \in \mathrm {GU}(V_{\Lambda }^{\epsilon })$ and $c(g_0) = c(g_1)$ . Here, $c(g_{\epsilon }) \in \mathbb F_{p}^{\times }$ denotes the multiplier of $g_{\epsilon }$ . For $0\leq \theta \leq \theta _{\mathrm {max}}$ and $\epsilon \in \{0,1\}$ , we will write $V_{\theta }^{\epsilon }$ and ${\mathcal {J}}_{\theta }$ instead of $V_{\Lambda _{\theta }}^{\epsilon }$ and ${\mathcal {J}}_{\Lambda _{\theta }}$ .

Let $\Lambda \in {\mathcal {L}}_i$ where $ni$ is even. We write $t(\Lambda ) = 2\theta +1$ . Let k be a perfect field extension of $\mathbb F_{p^2}$ . Let T be any $W(k)$ -lattice in $\mathbf V_k$ such that

$$ \begin{align*}p^{i+1}T^{\vee} \overset{2\theta'+1}{\subset} T \subset \Lambda_k\end{align*} $$

where $0 \leq \theta ' \leq \theta $ . Then T must contain $p^{i+1}\Lambda _k^{\vee }$ and $[\Lambda _k:T] = \theta - \theta '$ . We may consider $\overline {T} := T/p^{i+1}\Lambda _k^{\vee }$ the image of T in $V^{(0)}_{\Lambda }$ . Then $\overline {T}$ is an $\mathbb F_{p^2}$ -subspace of dimension $\theta + \theta ' + 1$ . Moreover, one may check that $\overline {p^{i+1}T^{\vee }} = \overline {T}^{\perp }$ , therefore the subspace $\overline {T}$ contains its orthogonal. These observations lead to the following proposition, see [Reference Vollaard32] Proposition 2.7.

Proposition 1.9. The mapping $T \mapsto \overline {T}$ defines a bijection between the set of $W(k)$ -lattices T in $\mathbf V_k$ such that $p^{i+1}T^{\vee } \overset {2\theta '+1}{\subset } T \subset \Lambda _k$ and the set

$$ \begin{align*}\{U\subset (V_{\Lambda}^0)_k \,|\, \dim U = \theta+\theta'+1 \text{ and }U^{\perp}\subset U\}.\end{align*} $$

In particular taking $\theta ' = 0$ , this set is in bijection with ${\mathcal {M}}_{\Lambda }(k)$ .

Remark 1.10. Similarly, the set of $W(k)$ -lattices T such that $\Lambda _k \subset T \overset {n-2\theta '-1}{\subset } p^{i}T^{\vee }$ for some $\theta \leq \theta ' \leq \theta _{\mathrm {max}}$ is in bijection with

$$ \begin{align*}\{U\subset (V_{\Lambda}^1)_k \,|\, \dim U = n - \theta' - \theta - 1 \text{ and }U^{\perp}\subset U\}.\end{align*} $$

The bijection is given by $T \mapsto \overline {T}^{\perp }$ where $\overline {T} := T/\Lambda _k \subset V^{(1)}_k$ . These sets can be seen as the k-rational points of some flag variety for $\mathrm {GU}(V^{(0)}_{\Lambda })$ and $\mathrm {GU}(V^{(1)}_{\Lambda })$ , which are special instances of Deligne-Lusztig varieties. This is accounted for in the next paragraph.

Let $\Lambda \in \mathcal L$ . The action of $J({\mathbb {Q}}_p)$ on the Rapoport-Zink space $\mathcal M$ restricts to an action of the parahoric subgroup $J_{\Lambda }$ on the closed Bruhat-Tits stratum ${\mathcal {M}}_{\Lambda }$ . This action factors through the maximal reductive quotient ${\mathcal {J}}_{\Lambda } \simeq \mathrm {G}(\mathrm {U}(V_{\Lambda }^0) \times \mathrm {U}(V_{\Lambda }^1))$ . This action is trivial on the normal subgroup $\{\mathrm {id}\}\times \mathrm U(V_{\Lambda }^1) \subset {\mathcal {J}}_{\Lambda }$ , thus it factors again through the quotient which is isomorphic to $\mathrm {GU}(V_{\Lambda }^0)$ . With respect to this action, the variety ${\mathcal {M}}_{\Lambda }$ is isomorphic to a generalized Deligne-Lusztig variety, see [Reference Vollaard and Wedhorn33] Theorem 4.8.

Theorem 1.11. There is an isomorphism between ${\mathcal {M}}_{\Lambda }$ and a certain generalized parabolic Deligne-Lusztig variety for the finite group of Lie type $\mathrm {GU}(V_{\Lambda }^{0})$ , compatible with the actions. In particular, if $t(\Lambda ) = 2\theta + 1$ then the variety ${\mathcal {M}}_{\Lambda }$ is projective, smooth, geometrically irreducible of dimension $\theta $ .

We refer to [Reference Muller25] Section 1 for the definition of Deligne-Lusztig varieties. In particular, the adjective “generalized” is understood according to loc. cit. The Deligne-Lusztig variety isomorphic to ${\mathcal {M}}_{\Lambda }$ is introduced in [Reference Vollaard and Wedhorn33] Section 4.5, and it is denoted by $Y_{\Lambda }$ there. Theorem 5.1 of loc. cit. describes the incidence relations between the different strata.

Theorem 1.12. Let $i\in \mathbb Z$ such that $ni$ is even. Let $\Lambda , \Lambda ' \in {\mathcal {L}}_i$ . The following statements hold.

1. (1) The inclusion $\Lambda \subset \Lambda '$ is equivalent to the scheme-theoretic inclusion ${\mathcal {M}}_{\Lambda }\subset {\mathcal {M}}_{\Lambda '}$ . It also implies $t(\Lambda )\leq t(\Lambda ')$ and there is equality if and only if $\Lambda = \Lambda '$ .

2. (2) The three following assertions are equivalent.

   $$ \begin{align*} \mathrm{(i)}\;\Lambda\cap \Lambda'\in {\mathcal{L}}_i. & & \mathrm{(ii)}\; \Lambda\cap \Lambda' \text{ contains a lattice of }{\mathcal{L}}_i. & & \mathrm{(iii)}\;{\mathcal{M}}_{\Lambda}\cap {\mathcal{M}}_{\Lambda'} \not = \emptyset. \end{align*} $$

   If these conditions are satisfied, then ${\mathcal {M}}_{\Lambda }\cap {\mathcal {M}}_{\Lambda '}={\mathcal {M}}_{\Lambda \cap \Lambda '}$ , where we understand the left-hand side as the scheme theoretic intersection inside ${\mathcal {M}}_{i,\mathrm {red}}$ .

3. (3) The three following assertions are equivalent

   $$ \begin{align*} & \mathrm{(i)}\;\Lambda+ \Lambda'\in {\mathcal{L}}_i. & & \mathrm{(ii)}\; \Lambda+ \Lambda' \text{ is contained in a lattice of }{\mathcal{L}}_i. & & \\ & \mathrm{(iii)}\;{\mathcal{M}}_{\Lambda}, {\mathcal{M}}_{\Lambda'} \subset {\mathcal{M}}_{\widetilde{\Lambda}} \text{ for some }\widetilde{\Lambda} \text{ in } {\mathcal{L}}_i. \end{align*} $$

   If these conditions are satisfied, then ${\mathcal {M}}_{\Lambda + \Lambda '}$ is the smallest subscheme of the form ${\mathcal {M}}_{\widetilde {\Lambda }}$ containing both ${\mathcal {M}}_{\Lambda }$ and ${\mathcal {M}}_{\Lambda '}$ .

4. (4) If k is a perfect field extension of $\mathbb F_{p^2}$ then ${\mathcal {M}}_i(k)=\bigcup _{\Lambda \in {\mathcal {L}}_i}{\mathcal {M}}_{\Lambda }(k)$ .

In essence, the previous statements explain how the stratification given by the ${\mathcal {M}}_{\Lambda }$ mimics the combinatorics of the Bruhat-Tits building of $\tilde J$ , hence the name.

1.3 Normalizers of maximal parahoric subgroups of $J({\mathbb {Q}}_p)$

In this section we compute the normalizer of the maximal parahoric subgroups $J_{\Lambda }$ .

Lemma 1.13. Let $\Lambda , \Lambda ' \in \mathcal L$ .

1. (i) The parahoric subgroup $J_{\Lambda }$ acts transitively on the set of apartments containing $\Lambda $ .

2. (ii) We have $J_{\Lambda } = J_{\Lambda '}$ if and only if there exists $k\in \mathbb Z$ such that $\Lambda = p^k\Lambda '$ or $\Lambda = p^k\Lambda '\,^{\vee }$ .

Proof. The first point is a general fact from the theory of Bruhat-Tits buildings. For the second point, the converse is clear. Indeed, if $x\in {\mathbb {Q}}_{p^2}^{\times }$ then $J_{x\Lambda } = J_{\Lambda }$ , and an element $g\in J({\mathbb {Q}}_p)$ fixes a lattice $\Lambda $ if and only if it fixes its dual $\Lambda ^{\vee }$ . Now, let $\Lambda , \Lambda ' \in \mathcal L$ such that $J_{\Lambda } = J_{\Lambda '}$ . Up to replacing $\Lambda '$ with an appropriate lattice $g\cdot \Lambda '$ , it is enough to treat the case $\Lambda ' = \Lambda _{\theta }$ for some $0\leq \theta \leq \theta _{\mathrm {max}}$ . By Proposition 1.5, we can find an apartment $\mathcal A^L$ containing both $\Lambda _{\theta }$ and $\Lambda $ . By the first point, we can find $g\in J_{\theta } = J_{\Lambda }$ which sends $\mathcal A^L$ to $\mathcal A^e$ . Therefore $g\cdot \Lambda = \Lambda $ belongs to $\mathcal A^e$ . According to Proposition 1.7, we may write

$$ \begin{align*}\Lambda = \Lambda(r_{-\theta_{\mathrm{max}}},\ldots ,r_{-1},s,r_1,\ldots ,r_{\theta_{\mathrm{max}}})\end{align*} $$

for some integers $(r_j,s)$ . Let i be the integer such that $\Lambda \in {\mathcal {L}}_i$ . Then according to Proposition 1.6 we have

1. – $\forall 1\leq j \leq \theta _{\mathrm {max}}, r_{-j} + r_{j} \in \{i,i+1\}$ .

2. – $s_0 = \lfloor \frac {i+1}{2} \rfloor $ .

3. – if n is even then $s_1 = \lfloor \frac {i}{2} \rfloor $ .

For $1\leq j\leq \theta $ , let $g_j$ be the automorphism of $\mathbf V$ which exchanges $e_{-j}$ and $e_j$ while fixing all the other vectors in the basis e. Then, from the definition of $\Lambda _{\theta }$ we have $g_j \in J_{\theta }$ . Therefore $g_j$ must fix $\Lambda $ too, which implies that $r_{-j} = r_j$ . And for $\theta + 1 \leq j \leq \theta _{\mathrm {max}}$ , let $g_j$ be the automorphism sending $e_{j}$ to $p^{-1}e_{-j}$ and $e_{-j}$ to $pe_j$ while fixing all the other vectors in the basis e. Then again we have $g_j \in J_{\theta } = J_{\Lambda }$ which implies that $r_{-j} = r_j - 1$ .

Assume first that $i = 2i'$ is even. Combining the previous observations, we have $r_j = i'$ for all $1\leq j \leq \theta $ and $r_j = i' + 1$ for all $\theta + 1 \leq j \leq \theta _{\mathrm {max}}$ . Moreover we have $s_0 = i'$ and if n is even, we have $s_1 = i'$ . In other words, we have $\Lambda = p^{i'}\Lambda _{\theta }$ .

Assume now that $i = 2i' + 1$ is odd. This implies that n is even. Combining the previous observations, we have $r_j = i' + 1$ for all $1\leq j \leq \theta _{\mathrm {max}}$ . Moreover we have $s_0 = i' + 1$ and if n is even, we have $s_1 = i'$ . In other words, we have $\Lambda = p^{i'+1}\Lambda _{\theta }^{\vee }$ .

Proposition 1.14. Let $\Lambda \in \mathcal L$ . If $t(\Lambda ) \not = n - t(\Lambda )$ then the normalizer of $J_{\Lambda }$ in $J({\mathbb {Q}}_p)$ is $\mathrm N_J(J_{\Lambda }) = \mathrm Z(J({\mathbb {Q}}_p))J_{\Lambda }$ . Otherwise, n is even and there exists an element $h_0 \in J({\mathbb {Q}}_p)$ such that $h_0^2 = p\cdot \mathrm {id}$ and $N_J(J_{\Lambda })$ is the subgroup generated by $J_{\Lambda }$ and $h_0$ . In particular, $\mathrm Z(J({\mathbb {Q}}_p))J_{\Lambda }$ is a subgroup of index $2$ in $N_J(J_{\Lambda })$ .

Remark 1.15. The condition $t(\Lambda ) \not = n - t(\Lambda )$ is automatically satisfied if n is odd. If n is even, it is satisfied when $t(\Lambda ) \not = \theta _{\mathrm {max}} + 1$ , this is the case in particular when $\theta _{\mathrm {max}}$ is odd.

Proof. It is clear that $\mathrm Z(J({\mathbb {Q}}_p))J_{\Lambda } \subset \mathrm N_J(J_{\Lambda })$ . Conversely, let $g \in N_J(J_{\Lambda })$ , so that we have $J_{\Lambda } = {}^gJ_{\Lambda } = J_{g\cdot \Lambda }$ . We apply Lemma 1.13 to deduce the existence of $k\in \mathbb Z$ such that $g\cdot \Lambda = p^{k} \Lambda $ (case $1$ ) or $g\cdot \Lambda = p^{k} \Lambda ^{\vee }$ (case $2$ ). If we are in case $1$ , then $g \in p^{k}J_{\Lambda } \subset \mathrm Z(J({\mathbb {Q}}_p))J_{\Lambda }$ and we are done. If n is even, the assumption that $t(\Lambda ) \not = n - t(\Lambda )$ makes case $2$ impossible. If n is odd and we are in case $2$ , then in particular $\Lambda ^{\vee } \in \mathcal L$ . By Lemma 1.3, we must have $\Lambda = p^{i}\Lambda ^{\vee }$ for some even $i\in \mathbb Z$ . In particular, we are also in case $1$ . Therefore, no matter the parity of n, we are always in case $1$ .

Assume now that $t(\Lambda ) = n - t(\Lambda )$ , in particular n and $\theta _{\mathrm {max}}$ are both even. We write $\theta _{\mathrm {max}} = 2\theta _{\mathrm {max}}'$ so that $t(\Lambda ) = 2\theta _{\mathrm {max}}'+1$ and we solve the case $\Lambda = \Lambda _{\theta _{\mathrm {max}}'}$ first. Recall the element $g_0$ that was defined earlier. By direct computation, we see that $g_0\cdot \Lambda _{\theta _{\mathrm {max}}'} = p\Lambda _{\theta _{\mathrm {max}}'}^{\vee }$ . Therefore ${}^{g_0}J_{\theta _{\mathrm {max}}'} = J_{p\Lambda _{\theta _{\mathrm {max}}'}^{\vee }} = J_{\theta _{\mathrm {max}}'}$ so that $g_0 \in {\mathrm {N}}_J(J_{\theta _{\mathrm {max}}'})$ . Now let g be any element normalizing $J_{\theta _{\mathrm {max}}}$ , so that $J_{\theta _{\mathrm {max}}'} = {}^gJ_{\theta _{\mathrm {max}}'} = J_{g\cdot \Lambda _{\theta _{\mathrm {max}}'}}$ . According to 1.13 there exists $k\in \mathbb Z$ such that $g\cdot \Lambda _{\theta _{\mathrm {max}}'} = p^k\Lambda _{\theta _{\mathrm {max}}'}$ or $g\cdot \Lambda _{\theta _{\mathrm {max}}'} = p^k\Lambda _{\theta _{\mathrm {max}}'}^{\vee } = p^{k-1}g_0\cdot \Lambda _{\theta _{\mathrm {max}}'}$ . In the first case we have $g \in p^kJ_{\theta _{\mathrm {max}}'}$ and in the second case we have $g \in p^{k-1}g_0J_{\theta _{\mathrm {max}}'}$ . Since $g_0^2 = p\cdot \mathrm {id}$ , the claim is proved with $h_0 = g_0$ .

In the general case, we have $t(\Lambda ) = 2\theta _{\mathrm {max}}'+1 = t(\Lambda _{\theta _{\mathrm {max}}'})$ . There exists some $g\in J({\mathbb {Q}}_p)$ such that $\Lambda = g\cdot \Lambda _{\theta _{\mathrm {max}}'}$ . Then ${\mathrm {N}}_J(\Lambda ) = {}^g{\mathrm {N}}_J(\Lambda _{\theta _{\mathrm {max}}'})$ so that the claim follows with $h_0 := gg_0g^{-1}$ .

1.4 Counting the closed Bruhat-Tits strata

In this section we count the number of closed Bruhat-Tits strata which contain or which are contained in another given one. Let $d\geq 0$ and consider V a d-dimensional $\mathbb F_{p^2}$ -vector space equipped with a non-degenerate hermitian form. This structure is uniquely determined up to isomorphism as we are working over a finite field. For $\left \lceil \frac {d}{2}\right \rceil \leq r \leq d$ , we define

$$ \begin{align*} N(r,V) & := \{U \,|\, U \text{ is an }r\text{-dimensional subspace of }V \text{ such that }U^{\perp}\subset U\},\\ \nu(r,d) & := \# N(r,V), \end{align*} $$

where $U^{\perp }$ denotes the orthogonal of U with respect to the hermitian form on V. By Proposition 1.9 and the following Remark, we have the following statement, see also [Reference Vollaard and Wedhorn33] Corollary 5.7.

Proposition 1.16. Let $\Lambda \in \mathcal L$ . Write $t(\Lambda ) = 2\theta +1$ for some $0\leq \theta \leq \theta _{\mathrm {max}}$ .

1. – Let $\theta '$ be an integer such that $0\leq \theta ' \leq \theta $ . The number of closed Bruhat-Tits strata of dimension $\theta '$ contained in ${\mathcal {M}}_{\Lambda }$ is $\nu (\theta + \theta ' + 1, 2\theta + 1)$ .

2. – Let $\theta '$ be an integer such that $\theta \leq \theta ' \leq \theta _{\mathrm {max}}$ . The number of closed Bruhat-Tits strata of dimension $\theta '$ containing ${\mathcal {M}}_{\Lambda }$ is $\nu (n - \theta - \theta ' - 1, n - 2\theta - 1)$ .

Another way to formulate the proposition is to say that $\nu (\theta + \theta ' + 1, 2\theta + 1)$ (resp. $\nu (n - \theta - \theta ' - 1, n - 2\theta - 1)$ ) is the number of vertices of type $2\theta ' + 1$ in the Bruhat-Tits building of $\tilde {J}$ which are neighbours of a given vertex of type $2\theta + 1$ for $\theta ' \leq \theta $ (resp. $\theta ' \geq \theta $ ). In [Reference Vollaard and Wedhorn33], an explicit formula is given for $\nu (d-1,d)$ . The next proposition gives a formula to compute $\nu (r,d)$ for general r and d.

Proposition 1.17. Let $d\geq 0$ and let $\left \lceil \frac {d}{2}\right \rceil \leq r \leq d$ . We have

$$ \begin{align*}\nu(r,d) = \frac{\prod_{j=1}^{2(d-r)}\left(p^{2r-d+j} - (-1)^{2r-d+j}\right)}{\prod_{j=1}^{d-r}\left(p^{2j} -1\right)}\end{align*} $$

Proof. We fix a basis $(e_1,\ldots ,e_d)$ of V in which the hermitian form is represented by the matrix $A_d$ . We denote by $U_0$ the subspace generated by the vectors $e_1,\ldots , e_r$ . The orthogonal of $U_0$ is generated by $e_1,\ldots , e_{d-r}$ . Since $\left \lceil \frac {d}{2}\right \rceil \leq r \leq d$ , $U_0$ contains its orthogonal, thus $U_0 \in N(r,V)$ . The unitary group $\mathrm {U}(V) \simeq \mathrm U_{d}(\mathbb F_p)$ acts transitively on the set $N(r,V)$ (since $p \not = 2$ ). The stabilizer of $U_0$ in $\mathrm U_{d}(\mathbb F_p)$ is the standard parabolic subgroup

$$ \begin{align*}P_0 := \left\{ \begin{pmatrix} B & * & *\\ 0 & M & *\\ 0 & 0 & F(B) \end{pmatrix} \in \mathrm U_{d}(\mathbb F_p) \; \middle| \; B \in \mathrm{GL}_{d-r}(\mathbb F_{p^2}), M\in \mathrm U_{2r-d}(\mathbb F_p)\right\}.\end{align*} $$

Here, $F(B) = A_{d-r}(B^{(p)})^{-T}A_{d-r}$ where $B^{(p)}$ is the matrix B with all coefficients raised to the power p. The order of $\mathrm U_d(\mathbb F_p)$ is well known and given by the formula

$$ \begin{align*}\#\mathrm U_d(\mathbb F_p) = p^{\frac{d(d-1)}{2}}\prod_{j=1}^{d} \left(p^j - (-1)^j\right).\end{align*} $$

It remains to compute the order of $P_0$ . We have a Levi decomposition $P_0 = L_0N_0$ with $L_0 \cap N_0 = \{1\}$ where

$$ \begin{align*} L_0 & := \left\{ \begin{pmatrix} B & 0 & 0\\ 0 & M & 0\\ 0 & 0 & F(B) \end{pmatrix} \in \mathrm U_{d}(\mathbb F_p) \; \middle| \; B \in \mathrm{GL}_{d-r}(\mathbb F_{p^2}), M\in \mathrm U_{2r-d}(\mathbb F_p)\right\}, \\ N_0 & := \!\left\{\! \!\begin{pmatrix} 1 & X & Z\\ 0 & 1 & Y\\ 0 & 0 & 1 \end{pmatrix} \!\!\in \mathrm U_{d}(\mathbb F_p) \;\! \!\middle|\!\! \; X \in \mathrm{M}_{d-r,2r-d}(\mathbb F_{p^2}), Y\in \mathrm M_{2r-d,d-r}(\mathbb F_{p^2}), Z\in \mathrm M_{d-r}(\mathbb F_{p^2})\right\}. \end{align*} $$

The order of $L_0$ is given by

$$ \begin{align*}\#L_0 &= \#\mathrm{GL}_{d-r}(\mathbb F_{p^2})\#\mathrm U_{2r-d}(\mathbb F_p)\\& = p^{(d-r)(d-r-1) + \frac{(2r-d)(2r-d-1)}{2}}\prod_{j=1}^{d-r}\left(p^{2j}-1\right)\prod_{j=1}^{2r-d} \left(p^j - (-1)^j\right).\end{align*} $$

As for $N_0$ , we need some more conditions on the matrices $X, Y$ and Z. By direct computations, one may check that

$$ \begin{align*} \begin{pmatrix} 1 & X & Z\\ 0 & 1 & Y\\ 0 & 0 & 1 \end{pmatrix} \in N_0 \iff Y &= -A_{2r-d}(X^{(p)})^TA_{d-r} \text{ and } Z + A_{d-r}(Z^{(p)})^TA_{d-r} \\&= XY \in \mathrm M_{d-r}(\mathbb F_{p^2}). \end{align*} $$

Thus, X is any matrix of size $(d-r) \times (2r-d)$ and Y is determined by X. In the second equation, the matrix $A_{d-r}(Z^{(p)})^TA_{d-r}$ is the reflection of $Z^{(p)}$ with respect to the antidiagonal. The equation implies that the coefficients below the antidiagonal of Z determine those above the antidiagonal. Furthermore, if z is a coefficient in the antidiagonal then the equation determines the value of $\mathrm {Tr}(z) = z + z^p$ , where $\mathrm {Tr}: \mathbb F_{p^2} \rightarrow \mathbb F_p$ is the trace relative to the extension $\mathbb F_{p^2}/\mathbb F_p$ . The trace is surjective and its kernel has order p. Thus, there are only p possibilities for each antidiagonal coefficient. Putting things together, the order of $N_0$ is given by

$$ \begin{align*}\#N_0 = p^{2(d-r)(2r-d)}\cdot p^{2\frac{(d-r)(d-r-1)}{2}}\cdot p^{d-r} = p^{(d-r)(3r-d)}\end{align*} $$

where the three terms take account, respectively, of the choice of X, the choice of the coefficients below the antidiagonal of Z and the choice of the coefficients in the antidiagonal of Z. Hence the order of $P_0$ is given by

$$ \begin{align*}\#P_0 = \#L_0\#N_0 = p^{\frac{d(d-1)}{2}}\prod_{j=1}^{d-r}\left(p^{2j}-1\right)\prod_{j=1}^{2r-d} \left(p^j - (-1)^j\right).\end{align*} $$

Upon taking the quotient $\nu (r,d) = \#\mathrm U_d(\mathbb F_p)/\#P_0$ , the result follows.

In particular with $r = d-1$ , we obtain

$$ \begin{align*}\nu(d-1,d) = \frac{(p^{d-1}-(-1)^{d-1})(p^d-(-1)^{d})}{p^2-1}.\end{align*} $$

If $d = 2\delta $ is even, it is equal to $(p^{d-1}+1)\sum _{j=0}^{\delta -1}p^{2j}$ , and if $d = 2\delta + 1$ is odd, it is equal to $(p^{d}+1)\sum _{j=0}^{\delta -1}p^{2j}$ . This coincides with the formula given in [Reference Vollaard and Wedhorn33] Example 5.6.

2 The cohomology of a closed Bruhat-Tits stratum

In [Reference Muller25], we computed the cohomology groups $\mathrm {H}^{\bullet }_c({\mathcal {M}}_{\Lambda }\otimes \mathbb F,\overline {{\mathbb {Q}}_{\ell }})$ of the closed Bruhat-Tits strata. The computation relies on the Ekedahl-Oort stratification on ${\mathcal {M}}_{\Lambda }$ which, in the language of Deligne-Lusztig varieties, translates into a stratification by Coxeter varieties for unitary groups of smaller sizes. The cohomology of Coxeter varieties is well known thanks to the work of Lusztig in [Reference Lusztig20]. In order to state our results, we recall the classification of unipotent representations of the finite unitary group.

Let q be a power of prime number p, and let $\mathbf G$ be a reductive connected group over an algebraic closure $\mathbb F$ of $\mathbb F_p$ . Assume that $\mathbf G$ is equipped with an $\mathbb F_q$ -structure induced by a Frobenius morphism F. Let $G = \mathbf G^F$ be the associated finite group of Lie type. Let $(\mathbf T,\mathbf B)$ be a pair consisting of an F-stable maximal torus $\mathbf T$ and an F-stable Borel subgroup $\mathbf B$ containing $\mathbf T$ . Let $\mathbf W = \mathbf W(\mathbf T)$ denote the Weyl group of $\mathbf G$ . The Frobenius F induces an action on $\mathbf W$ . For $w\in \mathbf W$ , let $\dot {w}$ be a representative of w in the normalizer $\mathrm N_{\mathbf G}(\mathbf T)$ of $\mathbf T$ . By the Lang-Steinberg theorem, one can find $g\in \mathbf G$ such that $\dot {w} = g^{-1}F(g)$ . Then ${}^g\mathbf T := g\mathbf T g^{-1}$ is another F-stable maximal torus, and $w \in \mathbf W$ is said to be the type of $^g \mathbf T$ with respect to $\mathbf T$ . Every F-stable maximal torus arises in this manner. According to [Reference Deligne and Lusztig12] Corollary 1.14, the G-conjugacy class of $^g \mathbf T$ only depends on the F-conjugacy class of w in the Weyl group $\mathbf W$ . Here, two elements w and $w'$ in $\mathbf W$ are said to be F-conjugate if there exists some element $\tau \in \mathbf W$ such that $w = \tau w' F(\tau )^{-1}$ . For every $w\in \mathbf W$ , we fix $\mathbf T_w$ an F-stable maximal torus of type w with respect to $\mathbf T$ . The Deligne-Lusztig induction of the trivial representation of $\mathbf T_w$ is the virtual representation of G defined by the formula

$$ \begin{align*}R_w := \sum_{i\geq 0} (-1)^i{\mathrm{H}}^i_c(X(w)\otimes \, \mathbb F,\overline{{\mathbb{Q}}_{\ell}}),\end{align*} $$

where $X(w)$ is the Deligne-Lusztig variety for $\mathbf G$ given by

$$ \begin{align*}X(w) := \{g\mathbf B \in \mathbf G/\mathbf B \,|\, g^{-1}F(g)\in \mathbf B w \mathbf B\}.\end{align*} $$

According to [Reference Deligne and Lusztig12] Theorem 1.6, the virtual representation $R_w$ only depends on the F-conjugacy class of w in $\mathbf W$ . An irreducible representation of G is said to be unipotent if it occurs in $R_w$ for some $w\in \mathbf W$ . The set of isomorphism classes of unipotent representations of G is denoted by $\mathcal E(G,1)$ .

Let $\mathbf G$ and $\mathbf G'$ be two reductive connected groups over $\mathbb F$ both equipped with an $\mathbb F_q$ -structure. We denote by F and $F'$ the respective Frobenius morphisms. Let $f: \mathbf G\rightarrow \mathbf G'$ be an $\mathbb F_q$ -isotypy, that is a homomorphism defined over $\mathbb F_q$ whose kernel is contained in the centre of $\mathbf G$ and whose image contains the derived subgroup of $\mathbf G'$ . Then, according to [Reference Digne and Michel13] Proposition 11.3.8, we have an equality

$$ \begin{align*}\mathcal E(G,1) = \{\rho\circ f \,|\, \rho \in \mathcal E(G',1)\}.\end{align*} $$

Thus, the irreducible unipotent representations of G and of $G'$ can be identified. We will use this observation in the case $G = \mathrm {U}_k(\mathbb F_q)$ and $G' = \mathrm {GU}_k(\mathbb F_q)$ . The corresponding reductive groups are $\mathbf G = \mathrm {GL}_k$ and $\mathbf G' = \mathrm {GL}_k \times \mathrm {GL}_1$ . The Frobenius morphisms can be defined as

$$ \begin{align*} F(M) = \dot{w_0}(M^{(q)})^{-T}\dot{w_0}, & & F'(M,c) = (c^q\dot{w_0}(M^{(q)})^{-T}\dot{w_0}, c^q). \end{align*} $$

Here, $\dot {w_0}$ is the $k\times k$ matrix with only $1$ ’s in the antidiagonal and $M^{(q)}$ is the matrix M whose entries are all raised to the power q. The isotypy $f: \mathbf G\rightarrow \mathbf G'$ is defined by $f(M) = (M,1)$ . It satisfies $F'\circ f = f \circ F$ , it is injective and its image contains the derived subgroup $\mathrm {SL}_n\times \{1\} \subset \mathbf G'$ . Hence, we obtain the following result.

Assume that the Coxeter graph of the reductive group $\mathbf G$ is a union of subgraphs of type $A_m$ (for various m). Let be the set of isomorphism classes of irreducible representations of its Weyl group $\mathbf W$ . The action of the Frobenius F on $\mathbf W$ induces an action on , and we consider the fixed point set . The following theorem of [Reference Lusztig and Srinivasan22] classifies the irreducible unipotent representations of G.

We recall how the bijection is constructed. According to loc. cit. if there is a unique automorphism $\widetilde {F}$ of V of finite order such that

$$ \begin{align*}R(V) := \frac{1}{|\mathbf W|}\sum_{w\in \mathbf W} \mathrm{Trace}(w\circ \widetilde{F} \,|\, V)R_w\end{align*} $$

is an irreducible representation of G. Then the map $V \mapsto R(V)$ is the desired bijection. In the case of $\mathrm U_k(\mathbb F_q)$ or $\mathrm {GU}_k(\mathbb F_q)$ , the Weyl group $\mathbf W$ is identified with the symmetric group $\mathfrak S_k$ and we have an equality . Moreover, the automorphism $\widetilde {F}$ is the multiplication by $w_0$ , where $w_0$ is the element of maximal length in $\mathbf W$ . Thus, in both cases the irreducible unipotent representations of G are classified by the irreducible representations of the Weyl group $\mathbf W\simeq \mathfrak S_k$ , which in turn are classified by partitions of k or equivalently by Young diagrams, as we briefly recall in the next paragraph.

A partition of k is a tuple of integers $\lambda = (\lambda _1 \geq \ldots \geq \lambda _r>0)$ with $r\geq 1$ such that $\lambda _1 + \ldots + \lambda _r = k$ . The integer k is called the length of the partition, and it is denoted by $|\lambda |$ . A Young diagram of size k is a top left justified collection of k boxes, arranged in rows and columns. There is a correspondence between Young diagrams of size k and partitions of k, by associating to a partition $\lambda = (\lambda _1, \ldots , \lambda _r)$ the Young diagram having r rows consisting successively of $\lambda _1, \ldots , \lambda _r$ boxes. We will often identify a partition with its Young diagram, and conversely. For example, the Young diagram associated to $\lambda = (3^2,2^2,1)$ is the following one.

To any partition $\lambda $ of k, one can naturally associate an irreducible character $\chi _{\lambda }$ of the symmetric group $\mathfrak S_k$ .

The irreducible unipotent representation of $\mathrm U_k(\mathbb F_q)$ (resp. $\mathrm {GU}_k(\mathbb F_q)$ ) associated to $\chi _{\lambda }$ by the bijection of Theorem 2.3 is denoted by $\rho _{\lambda }^{\mathrm U}$ (resp. $\rho _{\lambda }^{\mathrm {GU}}$ ). In virtue of Proposition 2.2, for every $\lambda $ we have $\rho _{\lambda }^{\mathrm U} = \rho _{\lambda }^{\mathrm {GU}} \circ f$ , where $f: \mathrm U_k(\mathbb F_q) \rightarrow \mathrm {GU}_k(\mathbb F_q)$ is the inclusion. Thus, it is harmless to identify $\rho _{\lambda }^{\mathrm U}$ and $\rho _{\lambda }^{\mathrm {GU}}$ so that from now on, we will omit the superscript. The partition $(k)$ corresponds to the trivial representation and $(1^k)$ to the Steinberg representation. Given a box in the Young diagram of $\lambda $ , its hook length is $1$ plus the number of boxes lying below it or on its right. For instance, in the following figure the hook length of every box of the Young diagram of $\lambda = (3^2,2^2,1)$ has been written inside it.

The degree of the representations $\rho _{\lambda }$ is given by expressions known as hook formula, see, for instance, [Reference Geck and Pfeiffer15] Proposition 4.3.5.

Proposition 2.4. Let $\lambda = (\lambda _1 \geq \ldots \geq \lambda _r> 0)$ be a partition of k. The degree of the irreducible unipotent representation $\rho _{\lambda }$ is given by the following formula

where $a(\lambda ) = \sum _{i=1}^r (i-1)\lambda _i$ .

We may describe the cuspidal support of the unipotent representations $\rho _{\lambda }$ . According to [Reference Lusztig21] Propositions 9.2 and 9.4 there exists an irreducible unipotent cuspidal representation of $\mathrm U_k(\mathbb F_q)$ (or $\mathrm {GU}_k(\mathbb F_q)$ ) if and only if k is an integer of the form $k = \frac {t(t+1)}{2}$ for some $t\geq 0$ . When k is an integer of this form, the unique unipotent cuspidal representation is associated to the partition $\Delta _t := (t, t-1,\ldots ,1)$ , whose Young diagram has the distinctive shape of a staircase. Here, as a convention $U_0(\mathbb F_q) = \mathrm {GU}_0(\mathbb F_q)$ denotes the trivial group. For example, here are the Young diagrams of $\Delta _1,\Delta _2$ and $\Delta _3$ . Of course, the one of $\Delta _0$ the empty diagram.

We consider an integer $t\geq 0$ such that k decomposes as $k = 2e + \frac {t(t+1)}{2}$ for some $e\geq 0$ . Let G denote $\mathrm {U}_k(\mathbb F_q)$ or $\mathrm {GU}_k(\mathbb F_q)$ , and consider $L_t$ the subgroup consisting of block-diagonal matrices having one middle block of size $\frac {t(t+1)}{2}$ and all other blocks of size $1$ . This is a standard Levi subgroup of G. For $\mathrm U_k(\mathbb F_q)$ , it is isomorphic to $\mathrm {GL}_1(\mathbb F_{q^2})^e\times \mathrm U_{\frac {t(t+1)}{2}}(\mathbb F_q)$ whereas in the case of $\mathrm {GU}_k(\mathbb F_q)$ it is isomorphic to $\mathrm {G}\left (\mathrm {U}_1(\mathbb F_{q})^e\times \mathrm {U}_{\frac {t(t+1)}{2}}(\mathbb F_q)\right )$ . In both cases, $L_t$ admits a quotient which is isomorphic to a group of the same type as G but of size $\frac {t(t+1)}{2}$ . We write $\rho _t$ for the inflation to $L_t$ of the unipotent cuspidal representation $\rho _{\Delta _t}$ of this quotient. If $\lambda $ is a partition of k, the cuspidal support of the representation $\rho _{\lambda }$ is given by exactly one of the pair $(L_t,\rho _t)$ up to conjugation, where $t\geq 0$ is an integer such that for some $e\geq 0$ we have $k = 2e + \frac {t(t+1)}{2}$ . Note that in particular k and $\frac {t(t+1)}{2}$ must have the same parity. With these notations, the irreducible unipotent representations belonging to the principal series (i.e. those whose cuspidal support is supported on a minimal parabolic subgroup) are those with cuspidal support $(L_0,\rho _0)$ if k is even and $(L_1,\rho _1)$ if k is odd.

Given an irreducible unipotent representation $\rho _{\lambda }$ , there is a combinatorical way to determine the Harish-Chandra series to which it belongs, as we recalled in [Reference Muller25] Section 2. We consider the Young diagram of $\lambda $ . We call domino any pair of adjacent boxes in the diagram. It may be either vertical or horizontal. We remove dominoes from the diagram of $\lambda $ so that the resulting shape is again a Young diagram, until one can not proceed further. This process results in the Young diagram of the partition $\Delta _t$ for some $t\geq 0$ , and it is called the $2$ -core of $\lambda $ . It does not depend on the successive choices for the dominoes. Then, the representation $\rho _{\lambda }$ has cuspidal support $(L_t,\rho _t)$ if and only if $\lambda $ has $2$ -core $\Delta _t$ . For instance, the diagram $\lambda = (3^2,2^2,1)$ has $2$ -core $\Delta _1$ , as it can be determined by the following steps. We put crosses inside the successive dominoes that we remove from the diagram.

Thus, the unipotent representation $\rho _{\lambda }$ of $\mathrm U_{11}(\mathbb F_q)$ or $\mathrm {GU}_{11}(\mathbb F_q)$ has cuspidal support $(L_1,\rho _1)$ , so in particular it is a principal series representation.

From now on, we take $q = p$ . Let $\Lambda \in \mathcal L$ with orbit type $t(\Lambda ) = 2\theta + 1$ . Recall that the stratum ${\mathcal {M}}_{\Lambda }$ is equipped with an action of the finite group of Lie type $\mathrm {GU}(V_{\Lambda }^0)$ . Upon choosing a basis, we identify this group with $\mathrm {GU}_{2\theta +1}(\mathbb F_p)$ . Let $\mathrm {Frob} = \sigma ^{-2} \in \mathrm {Gal}(\mathbb F/\mathbb F_{p^2})$ be the geometric Frobenius. Then $\mathrm {Frob}$ is a topological generator of $\mathrm {Gal}(\mathbb F/\mathbb F_{p^2})$ . In [Reference Muller25], we computed the cohomology groups $\mathrm {H}^{\bullet }({\mathcal {M}}_{\Lambda }\otimes \mathbb F,\overline {{\mathbb {Q}}_{\ell }})$ in terms of a $\mathrm {GU}_{2\theta +1}(\mathbb F_p) \times \langle \mathrm {Frob} \rangle $ -representation. The result is summed up in the following Theorem.

Thus, the cohomology of ${\mathcal {M}}_{\Lambda }$ consists only of unipotent representations whose associated Young diagram has at most two rows.

3 Shimura variety and p-adic uniformization of the supersingular locus

In this section, we introduce the PEL unitary Shimura variety with signature $(1,n-1)$ as in [Reference Vollaard and Wedhorn33] Sections 6.1 and 6.2, and we recall the p-adic uniformization theorem of its basic (or supersingular) locus. The Shimura variety can be defined as a moduli problem classifying abelian varieties with additional structures, as follows. Let $\mathbb E$ be a quadratic imaginary extension of ${\mathbb {Q}}$ such that $\mathbb E_p \simeq E$ . In particular p is inert in $\mathbb E$ . Let $B/\mathbb E$ be a simple central algebra of degree $d\geq 1$ which splits over p and at infinity. Let $*$ be a positive involution of the second kind on B, and let $\mathbb V$ be a non-zero finitely generated left B-module equipped with a non-degenerate $*$ -alternating form $\langle \cdot ,\cdot \rangle $ taking values in ${\mathbb {Q}}$ . Assume also that $\dim _{\mathbb E}(\mathbb V) = nd$ . Let $\mathbb G$ be the connected reductive group over ${\mathbb {Q}}$ whose points over a ${\mathbb {Q}}$ -algebra R are given by

$$ \begin{align*}\mathbb G(R) := \{g \in \mathrm{GL}_{\mathbb E\otimes R}(\mathbb V\otimes R) \,|\, \exists c\in R^{\times} \text{ such that for all } v,w \in \mathbb V\otimes R, \langle gv,gw\rangle = c\langle v,w\rangle \}.\end{align*} $$

We denote by $c:\mathbb G\rightarrow \mathbb G_{m}$ the multiplier character. The base change $\mathbb G_{\mathbb R}$ is isomorphic to a group of unitary similitudes $\mathrm {GU}(r,s)$ of a hermitian space with signature $(r,s)$ where $r+s=n$ . We assume that $r=1$ and $s=n-1$ . We consider a Shimura datum of the form $(\mathbb G,X)$ , where X denotes the unique $\mathbb G(\mathbb R)$ -conjugacy class of homomorphisms $h:\mathbb C^{\times } \rightarrow \mathbb G_{\mathbb R}$ such that for all $z\in \mathbb C^{\times }$ we have $\langle h(z)\cdot ,\cdot \rangle = \langle \cdot , h(\overline {z})\cdot \rangle $ , and such that the $\mathbb R$ -pairing $\langle \cdot ,h(i)\cdot \rangle $ is positive definite. Such a homomorphism h induces a decomposition $\mathbb V\otimes \mathbb C = \mathbb V_1 \oplus \mathbb V_2$ . Concretely, $\mathbb V_1$ (resp. $\mathbb V_2$ ) is the subspace where $h(z)$ acts like z (resp. like $\overline {z}$ ). Let F be the unique subfield of $\mathbb C$ isomorphic to $\mathbb E$ . The reflex field associated to the PEL data, that is the field of definition of $\mathbb V_1$ as a complex representation of B, is equal to F unless $n=2$ , in which case it is ${\mathbb {Q}}$ . Nonetheless, for simplicity we will consider the associated Shimura varieties over F even in the case $n=2$ .

Let $\mathbb A_f$ denote the ring of finite adèles over ${\mathbb {Q}}$ and let $K\subset G(\mathbb A_f)$ be an open compact subgroup. We define a functor $\mathrm {Sh}_K$ by associating to an F-scheme S the set of isomorphism classes of tuples $(A,\lambda _A,\iota _A,\overline {\eta }_A)$ where

1. – A is an abelian scheme over S.

2. – $\lambda _A: A\rightarrow \widehat {A}$ is a polarization.

3. – $\iota _A:B\rightarrow \mathrm {End}(A)\otimes {\mathbb {Q}}$ is a morphism of algebras such that $\iota _A(b^*) = \iota _A(b)^{\dagger }$ where $\cdot ^{\dagger }$ denotes the Rosati involution associated to $\lambda _A$ , and such that the Kottwitz determinant condition is satisfied:

   $$ \begin{align*}\forall b \in B,\, \det(\iota_A(b)) = \det(b\,|\, \mathbb V_1).\end{align*} $$

4. – $\overline {\eta }_A$ is a K-level structure, that is a K-orbit of isomorphisms of $B\otimes \mathbb A_f$ -modules $\mathrm {H}_1(A,\mathbb A_f) \xrightarrow {\sim } \mathbb V\otimes \mathbb A_f$ that is compatible with the other data.

The Kottwitz condition in the third point is independent of the choice of $h\in X$ . If K is sufficiently small, this moduli problem is represented by a smooth quasi-projective scheme $\mathrm {Sh}_K$ over F. When the level K varies, the Shimura varieties form a projective system $(\mathrm {Sh}_K)_K$ equipped with an action of $\mathbb G(\mathbb A_f)$ by Hecke correspondences.

We assume the existence of a $\mathbb Z_{(p)}$ -order $\mathcal O_B$ in B, stable under the involution $*$ , such that its p-adic completion is a maximal order in $B_{{\mathbb {Q}}_p}$ . We also assume that there is a $\mathbb Z_p$ -lattice $\Gamma $ in $\mathbb V\otimes {\mathbb {Q}}_p$ , invariant under $\mathcal O_B$ and self-dual for $\langle \cdot ,\cdot \rangle $ . We may fix isomorphisms $\mathbb E_p \simeq E$ and $B_{{\mathbb {Q}}_p} \simeq \mathrm M_d(E)$ such that $\mathcal O_B\otimes \mathbb Z_{p}$ is identified with $\mathrm M_d(\mathcal O_E)$ .

As a consequence of the existence of $\Gamma $ , the group $G := \mathbb G_{{\mathbb {Q}}_p}$ is unramified. Let $K_0 := \mathrm {Stab}(\Gamma )$ be the subgroup of $G({\mathbb {Q}}_p)$ consisting of all g such that $g\cdot \Gamma = \Gamma $ . It is a hyperspecial maximal compact subgroup of $G({\mathbb {Q}}_p)$ . We will consider levels of the form $K = K_0K^p$ where $K^p$ is an open compact subgroup of $\mathbb G(\mathbb A_f^p)$ . Note that K is sufficiently small as soon as $K^p$ is sufficiently small. By the work of Kottwitz in [Reference Kottwitz19], the Shimura varieties $\mathrm {Sh}_{K_0K^p}$ admit integral models over $\mathcal O_{F,(p)}$ which have the following moduli interpretation. We define a functor $\mathrm {S}_{K^p}$ by associating to an $\mathcal O_{F,(p)}$ -scheme S the set of isomorphism classes of tuples $(A,\lambda _A,\iota _A,\overline {\eta }^p_A)$ where

1. – A is an abelian scheme over S.

2. – $\lambda _A: A\rightarrow \widehat {A}$ is a polarization whose order is prime to p.

3. – $\iota _A:\mathcal O_B\rightarrow \mathrm {End}(A)\otimes \mathbb Z_{(p)}$ is a morphism of algebras such that $\iota _A(b^*) = \iota _A(b)^{\dagger }$ where $\cdot ^{\dagger }$ denotes the Rosati involution associated to $\lambda _A$ , and such that the Kottwitz determinant condition is satisfied:

   $$ \begin{align*}\forall b \in \mathcal O_B,\, \det(\iota_A(b)) = \det(b\,|\, \mathbb V_1).\end{align*} $$

4. – $\overline {\eta }^p_A$ is a $K^p$ -level structure, that is a $K^p$ -orbit of isomorphisms of $B\otimes \mathbb A_f^p$ -modules $\mathrm {H}_1(A,\mathbb A_f^p) \xrightarrow {\sim } \mathbb V\otimes \mathbb A_f^p$ that is compatible with the other data.

If $K^p$ is sufficiently small, this moduli problem is also representable by a smooth quasi-projective scheme over $\mathcal O_{F,(p)}$ . When the level $K^p$ varies, these integral Shimura varieties form a projective system $(\mathrm {S}_{K^p})_{K^p}$ equipped with an action of $\mathbb G(\mathbb A_f^p)$ by Hecke correspondences. We have a family of isomorphisms

$$ \begin{align*}\mathrm{Sh}_{K_0K^p} \simeq \mathrm{S}_{K^p}\otimes_{\mathcal O_{F,(p)}} F\end{align*} $$

which are compatible as the level $K^p$ varies.

Therefore, under this convention we have isomorphisms $\mathrm {Sh}_{K_0K^p}\otimes _F {\mathbb {Q}}_{p^2} \simeq \mathrm {S_{K^p}}\otimes _{\mathbb Z_{p^2}} {\mathbb {Q}}_{p^2}$ compatible as the level $K^p$ varies. Let $\overline {\mathrm S}_{K^p} := \mathrm S_{K^p}\otimes _{\mathbb Z_{p^2}} \mathbb F_{p^2}$ denote the special fiber of the Shimura variety. Let $\overline {\mathrm S}_{K^p}^{\mathrm {ss}}$ denote the supersingular locus of the Shimura variety, i.e. the locus of points $x \in \overline {\mathrm S}_{K^p}$ such that the universal abelian scheme is supersingular at x. Then $\overline {\mathrm S}_{K^p}^{\mathrm {ss}}$ is a closed subvariety of $\overline {\mathrm S}_{K^p}$ , and its geometry can be described using the Rapoport-Zink space $\mathcal M$ in a process called p-adic uniformization, see [Reference Rapoport and Zink28] and [Reference Fargues14].

Let $x = [{\mathcal {A}}_x,\lambda _x,\iota _x,\overline {\eta }^p_x]$ be a geometric point of $\overline {\mathrm S}_{K^p}^{\mathrm {ss}}$ . Since $\mathbb G$ satisfies the Hasse principle, according to [Reference Fargues14] Proposition 3.1.8 the isogeny class of $({\mathcal {A}}_x, \lambda _x, \iota _x)$ does not depend on the choice of x. The p-divisible group ${\mathcal {A}}_x[p^{\infty }]$ inherits an $\mathcal O_B \otimes \mathbb Z_p \simeq \mathrm M_d(\mathcal O_E)$ -action from $\iota _A$ . Let $\mathbb X_x := \mathcal O_E^d \otimes _{\mathrm M_d(\mathcal O_E)} {\mathcal {A}}_x[p^{\infty }]$ with $\mathcal O_E$ -action induced by the diagonal inclusion $\mathcal O_E \hookrightarrow \mathcal O_E^d$ . According to [Reference Vollaard and Wedhorn33] Section 6.3, $\mathbb X_x$ is a unitary p-divisible group of signature $(1,n-1)$ over $\mathbb F$ in the sense of Section 1. Let also ${\mathcal {M}}_x$ be the Rapoport-Zink space defined as in Section 1, but using $\mathbb X_x$ as a framing object. In particular ${\mathcal {M}}_x$ is a formal scheme over $\mathrm {Spf}(W(\mathbb F))$ . There exists an isogeny $\mathbb X \otimes \mathbb F \to \mathbb X_x$ of unitary p-divisible group, inducing an isomorphism ${\mathcal {M}}_{W(\mathbb F)} := \mathcal M \otimes _{\mathbb Z_{p^2}} W(\mathbb F) \xrightarrow {\sim } {\mathcal {M}}_x$ , see [Reference Vollaard and Wedhorn33] Section 6.4. The Rapoport-Zink space ${\mathcal {M}}_x$ is equipped with an action of the group $J_x({\mathbb {Q}}_p)$ where $J_x$ is the group of quasi-isogenies of the unitary p-divisible group $\mathbb X_x$ . The quasi-isogeny $\mathbb X \otimes \mathbb F \to \mathbb X_x$ identifies $J_x$ with J and makes the isomorphism between the Rapoport-Zink spaces $J({\mathbb {Q}}_p)$ -equivariant. We define $I := \mathrm {Aut}({\mathcal {A}}_x,\lambda _x,\iota _x)$ as a reductive group over ${\mathbb {Q}}$ . Since x is in the supersingular locus, the group I is the inner form of $\mathbb G$ such that $I_{{\mathbb {Q}}_p} = J$ (in fact $J_x$ , which is identified with J), $I_{\mathbb A_f} = \mathbb G_{\mathbb A_f^p}$ and $I(\mathbb R) \simeq \mathrm {GU}(0,n)$ , which is the unique inner form of $G(\mathbb R)$ that is compact modulo centre. In particular, one can think of $I({\mathbb {Q}})$ as a subgroup both of $J({\mathbb {Q}}_p)$ and of $G(\mathbb A_f^p)$ . Let $(\widehat {\mathrm S}_{K^p})^{\mathrm {ss}}$ denote the formal completion of $\mathrm S_{K^p}$ along the supersingular locus. The p-adic uniformization theorem relates $(\widehat {\mathrm S}_{K^p})^{\mathrm {ss}}$ with a certain quotient of ${\mathcal {M}}_x$ , see [Reference Rapoport and Zink28] Theorem 6.23. Using the isomorphism above, we may replace ${\mathcal {M}}_x$ with ${\mathcal {M}}_{W(\mathbb F)}$ and obtain the following statement.