0 Introduction

Let $\textbf{k}$ be an algebraically closed field of prime characteristic $p$, $G$ be a connected reductive algebraic group over $\textbf{k}$, and ${{\frak g}} = {\rm Lie}(G)$. Let $T$ be a maximal torus of $G$ and $X(T)$ the character group of $T$. Assume that the derived group $G^{(1)}$ of $G$ is simply connected, $p$ is a good prime for the root system of ${\frak g}$, and ${\frak g}$ has a non-degenerated $G$-invariant bilinear form. Associated with a given linear form $\chi\in{\frak g}^*$, the reduced enveloping algebra $U_{\chi}({\frak g})$ is defined to be the quotient of the universal enveloping algebra $U({\frak g})$ by the ideal generated by all $x^p-x^{[p]}-\chi(x)^p$ with $x\in {\frak g}$. Each isomorphism class of irreducible representations of ${\frak g}$ corresponds to a unique $p$-character $\chi\in{\frak g}^*$. Furthermore, a well-known result shows that there is a Morita equivalence between $U_{\chi}({\frak g})$-module category and $U_{\chi}({\frak l})$-module category, where ${\frak l}$ is a certain reductive subalgebra of ${\frak g}$^[1-2]. Thus, we can consider the representations of $U_{\chi}({\frak g})$ with just nilpotent $\chi$.

It is well-known that simple ${\frak g}$-modules with a regular nilpotent $p$-character $\chi$ can be realized as baby Verma modules. In the case where $\chi$ lies in the subregular nilpotent orbit and ${\frak g}={\frak {sl}}_{n+1}(\textbf{k})$ or $({\frak {so}})_{2n+1}(\textbf{k})$, the $p$-character $\chi$ is conjugate to a $p$-character of standard Levi form, the graded representation categories $(U_\chi({\frak g}), T_0)$-mod of ${\frak g}$ associated with certain torus corresponding to $\chi$ were studied thoroughly by Jantzen^[3]; for example, each baby Verma module $Z_\chi(\lambda)$ is uniserial and gave all the composition factors, where $T_0$ is a certain torus of $\mbox{SL}(n+1)$ associated with $\chi$.

In this paper, we present an interesting observation on hom-spaces between any two baby Verma modules in any given block. Let ${\frak g}$ be of type $A_n$ with a subregular nilpotent $p$-character $\chi$ where $p\nmid n+1$. We obtain the following results.

1. In Section 6, as graded $U_{\chi}({\frak g})$ modules, we take into account the hom-spaces between the adjacent baby Verma modules in the sequence $\widehat Z_\chi(\lambda_k), k = 0, 1, \cdots, n$. We give sufficient conditions for these hom-spaces being nonzero.

2. In Section 7, as $U_{\chi}({\frak g})$ modules, we consider the hom-spaces between $Z_\chi(\lambda_i)$ and $Z_\chi(\lambda_k)$ for all $i, j \in\{0, 1, \cdots, n\}$. We firstly get the hom-spaces between $Z_\chi(\lambda_0)$ and $Z_\chi(\lambda_i)$, then prove that $\hbox{Hom}_{U_{\chi}({\frak g})} (Z_\chi(\lambda_0), Z_\chi(\lambda_i)) $ and $\hbox{Hom}_{U_{\chi}({\frak g})} (Z_\chi(\lambda_i), Z_\chi(\lambda_0))$ are nonzero for the weight $\lambda_0$ which lies at arbitrary sites in the closure of the first dominant alcove. As a consequence, we obtain the result that $\hbox{Hom}_{U_{\chi}({\frak g})} (Z_\chi(\lambda_i), Z_\chi(\lambda_j))$ are nonzero for all $i, j \in\{0, 1, \cdots, n\}$; this ultimately reveals the strong linkage relationship between any two baby Verma modules in the same block (Theorem 7.6).

1 Preliminaries and Notations

The following notations are standardized:

(1) $G$, ${\frak g}$: connected reductive group over an algebraically closed field $\textbf{k}$ of prime characteristic $p$, the Lie algebra of $G$.

(2) $T$, $X(T)$: a maximal torus of $G$, the character group of $T$.

(3) $\chi$: $p$-character of ${\frak g}$.

(4) $U({\frak g})$, $U_\chi({\frak g})$: the universal enveloping algebra of ${\frak g}$, the reduced enveloping algebra $U_\chi({\frak g})$ of ${\frak g}$, i.e., $U_\chi({\frak g})=U({\frak g})/( x^{p}-x^{[p]}-\chi {(x)^p} ~\mid~ x\in {\frak g} )$.

(5) ${\frak g}={\frak n}^-\oplus{\frak h}\oplus{\frak n}^+$, ${\frak h}$, ${\frak b}^+$: the triangular decomposition of ${\frak g}$, Cartan subalgebra of ${\frak g}$, Borel subalgebra of ${\frak g}$ generated by all positive root vectors and ${\frak h}$.

(6) $\Pi$, $I$: the set of simple roots, a subset of $\Pi$.

(7) $R$, $R^+$, $R_I$, $R_I^+$: root system of ${\frak g}$, set of positive roots, root system of ${\frak g}_I$, set of positive roots in ${\frak g}_I$.

(8) $E$: Euclidean space spanned by $R$ with inner product $\langle, \rangle$.

(9) $\rho, \alpha ^{\vee}$: half the sum of positive roots, the coroot of $\alpha$.

(10) $s_\alpha$, $s_{\alpha, rp}$: reflection with $s_{\alpha}(\mu)=\mu-\langle\mu, \alpha^{\vee}\rangle\alpha$, affine reflection with $s_{\alpha, rp}(\mu)=\mu-(\langle\mu, \alpha^{\vee}\rangle-rp)\alpha$.

(11) $W$, $W_p$: the Weyl group generated by $\{s_\alpha\mid\alpha \in R \}$, the affine Weyl group generated by $\{s_{\alpha, rp}\mid\alpha\in R, r\in\mathbb{Z}\}$.

(12) $w.\lambda=w(\lambda+\rho)-\rho, w\in W$: the dot action of $w$ on $\lambda$.

(13) $C_0=\{\lambda\in X(T)\mid 0\leqslant\langle\lambda+\rho, \alpha^{\vee}\rangle< p \mbox{ for all }\alpha\in R^+ \}$: the first dominant alcove of $X(T)$ for the action of $W_p$.

2 Baby Verma modules

For $\chi\in{\frak g}^*$, set $\Lambda_{\chi}=\{\lambda\in {\frak h}^{*}\mid\lambda(h)^p-\lambda(h^{[p]})=\chi(h)^p, ~\forall~ h\in {\frak h}\}$. When $\chi$ is nilpotent, $\Lambda_{\chi}=\Lambda_{0}:=\{\lambda \in {\frak h}^*\mid \lambda(h)^p= \lambda(h^{[p]})\}\cong X(T)/ pX(T)$ ^[4]. In this paper, we always assume that $\chi({\frak b}^+)=0$, i.e., $\chi$ is nilpotent.

The induced module ${Z}_{\chi}(\lambda):=U_{\chi}({\frak g})\otimes_{U_{0}({\frak b}^+)} \textbf{k}_{\lambda}$ is called a baby Verma module. Accordingly, each simple $U_{\chi}({\frak g})$-module is the homomorphic image of some baby Verma module ${Z}_{\chi}(\lambda)$ for $\lambda\in\Lambda_0$ ^[4].

3 Standard Levi forms

We say that a $p$-character $\chi$ has standard Levi form if $\chi({\frak b}^+)=0$ and if there exists a subset $I$ of $\Pi$ such that $\chi ({\frak g} _{-\alpha})\neq 0$ for $\alpha \in{I} $ and $\chi ({\frak g} _{-\alpha})= 0$ for $\alpha \in{R^+ \backslash I}$ ^[4].

Let $W_I$ and $W_{I, p}$ be the Weyl group generated by $\{s_{\alpha}~\mid~\alpha\in R_I\}$ and the affine Weyl group generated by $\{s_{\alpha, rp}~\mid~\alpha\in R_I, r\in{\mathbb{Z}}\}$, respectively. In this case, ${Z}_{\chi}(\lambda)$ is indecomposable^[4]. In particular, the simple modules can be understood as below.

Lemma 3.1    Any simple modules are the unique simple quotients of $Z_\chi(\nu)$, denoted by $L_\chi(\nu)$ for $\nu\in \Lambda_0$.

4 Graded $U_\chi({\frak g})$-modules

Let $\chi$ be of standard Levi-form as above. An $X(T)/\mathbb ZI$-graded $\mathit{\boldsymbol{U}}_\chi({\frak g})$-module category $\mathcal C$ was defined by Jantzen (cf. [4]). For $\lambda\in X(T)$, let $\widehat{{L}}_{\chi}(\lambda)$ and $\widehat{{Z}}_{\chi}(\lambda)$ be the simple module and baby Verma module in $\mathcal C$, respectively.

Lemma 4.1^[4]    Let $\lambda, \mu\in X(T)$. Then,

$\widehat{{L}}_{\chi}(\lambda)\simeq\widehat{{L}}_{\chi}(\mu)\Leftrightarrow\widehat{{Z}}_{\chi}(\lambda) \simeq\widehat{{Z}}_{\chi}(\mu)\Leftrightarrow\mu\in W_{I, p}\cdot\lambda.$

In this paper, denote ${{Z}}_{\chi}(\lambda)$ by the non-graded $U_\chi({\frak g})$-module and $\widehat{{Z}}_{\chi}(\lambda)$ by the graded $U_\chi({\frak g})$-module.

5 Subregular representations of type $\mathit{\boldsymbol{A}}_\mathit{\boldsymbol{n}}$

In the rest of the paper, we always assume that ${\frak g}$ is of type $A_n$ with $p\nmid n+1$ and $\chi\in{\frak g}^*$ has standard Levi form associated with $I=\{\alpha\in\Pi\mid \chi({\frak g}_{-\alpha})\neq 0\}=\{\alpha_1, \alpha_2, \cdots, \alpha_{n-1}\}$. Let root system $R=\{\epsilon_i-\epsilon_j\, |\, 1\leqslant i\neq j\leqslant n+1\}$ where $\epsilon_i(t)=t_i, t\in T.$ Denote $\{\alpha_i=\epsilon_i-\epsilon_{i+1}\, |\, 1\leqslant i\leqslant n\}$ by the set of simple roots of $R$. The fundamental weights is denoted by $\varpi_1, \varpi_2, \cdots, \varpi_n.$

Set $\sigma=s_1s_2\cdot\cdot\cdot s_n$ where $s_i$ is the simple reflection corresponding to $\alpha_i$. Assume $\lambda_0\in C_0$ with $\lambda_0+\rho=(r_1, r_2, \cdots, r_n)$. Then $0\leqslant\sum\limits_{i=1}^nr_i\leqslant p$. Set $\lambda_i=\sigma^i.\lambda_0$ for $1\leqslant i\leqslant n$. Then $\lambda_i+\rho=(r_{n-i+2}, r_{n-i+3}, \cdots, r_n, -(r_1+\cdot\cdot\cdot +r_i), r_1, r_2, \cdots, r_{n-i})$.

Let $T_0=\bigcap_{i=1}^{n-1}\hbox{ker}(\alpha_i)$. Then the category of $U_\chi({\frak g})$-$T_0$ modules is isomorphic to the graded $U_\chi({\frak g})$ modules category in Section 4.

Each $\widehat L_\chi(w.\lambda_0)$ with $w\in W$ is isomorphic to some $\widehat L_\chi(\lambda_i)$ with $0\leqslant i\leqslant n$^[4]. Then $\{\widehat L_\chi(\lambda_i)\mid 0\leqslant i\leqslant n\}$ is the set of isomorphism classes of simple modules in the block containing $\widehat L_\chi(\lambda_0)$. It follows from the results in [4] that each $\widehat Z_\chi(\lambda_i)$ is uniserial and the composition factors (from top to bottom) are:

$ \widehat L_\chi(\lambda_i), \widehat L_\chi(\lambda_{i+1}), \cdots, \widehat L_\chi(\lambda_n), \widehat L_\chi(\lambda_{0})\otimes(-p(n+1)), \cdots, \widehat L_\chi(\lambda_{i-1})\otimes(-p(n+1)). $

6 Hom-spaces of graded baby Verma modules

Firstly, consider the Hom-spaces $\widehat Z_\chi(\lambda_{i+1})\longrightarrow\widehat Z_\chi(\lambda_i)$.

Lemma 6.1    Assume that $0\leqslant i <n, 0\leqslant r_{n-i}<p$. Then

$\mbox{Hom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_{i+1}), \widehat Z_\chi(\lambda_i))\neq0 .$

Proof    Because $\lambda_i=\sigma^i.\lambda_0$ for $1\leqslant i\leqslant n$, by Lemma $4.1$, we have $\widehat Z_\chi(\lambda_{i+1})=\widehat Z_\chi(\sigma.\lambda_i)\cong \widehat Z_\chi(s_n.\lambda_i)$. Furthermore, $s_n.\lambda_i=\lambda_i-\langle\lambda+\rho, \alpha_n^{\vee}\rangle\alpha_n=\lambda_i-r_{n-i}\alpha_n.$ Thus, $x_{-\alpha_n}^{r_{n-i}}\otimes v_{\lambda_i}$ is a maximal weight vector of $\widehat Z_\chi(\lambda_i)$ which means $x_\alpha.x_{-\alpha_n}^{r_{n-i}}\otimes v_{\lambda_i}=0, \forall \alpha\in R^+$.

Then we can get a nonzero homomorphism $\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n)\longrightarrow\widehat Z_\chi(\lambda_i)$ mapping $v_{\lambda_i-r_{n-i}\alpha_n}$ to $x_{-\alpha_n}^{r_{n-i}}\otimes v_{\lambda_i}$. The lemma has been proven.

Lemma 6.2    Assume that $0\leqslant i <n, 0\leqslant r_{n-i}<p$. Then

$\hbox{dimHom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_{i+1}), \widehat Z_\chi(\lambda_i))=1.$

Proof    Denote all the positive roots of ${\frak g}$ by $\beta_1, \beta_2, \cdots, \beta_r$. We know each basis element of $\widehat Z_\chi(\lambda)$ is of the form $\prod\limits^{r}_{i=1}x_{-\beta_i}^{a(\beta_i)}\otimes v_\lambda$.

By Lemma 6.1, let $f\in\mbox{Hom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n), \widehat Z_\chi(\lambda_i))$. Then, we get

$ \begin{align*} f: \widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n)&\longrightarrow\widehat Z_\chi(\lambda_i)\\ v_{\lambda_i-r_{n-i}\alpha_n}&\mapsto\sum\limits_{(\textbf{a})} \textbf{k}_\textbf{a}x_{-\beta_1}^{a(\beta_1)}\cdot\cdot\cdot x_{-\beta_r}^{a(\beta_r)}\otimes v_{\lambda_i}, \textbf{a}=(a(\beta_1), \cdots, a(\beta_r)). \end{align*} $

For each $h\in{\frak h}$, we have $f(h.v_{\lambda_i-r_{n-i}\alpha_n})=h.f(v_{\lambda_i-r_{n-i}\alpha_n})$. Furthermore, we have

$ \begin{align*}f(h.v_{\lambda_i-r_{n-i}\alpha_n})&=(\lambda_i-r_{n-i} \alpha_n)(h)f(v_{\lambda_i-r_{n-i}\alpha_n}) \\ &=(\lambda_i-r_{n-i}\alpha_n)(h)\sum\limits_{(\textbf{a})} \textbf{k}_\textbf{a}x_{-\beta_1}^{a(\beta_1)}\cdots x_{-\beta_r}^{a(\beta_r)}\otimes v_{\lambda_i}, \end{align*} $

and

$ \begin{align*} h.f(v_{\lambda_i-r_{n-i}\alpha_n})&=h.\sum\limits_{(\textbf{a})} \textbf{k}_\textbf{a}x_{-\beta_1}^{a(\beta_1)}\cdots x_{-\beta_r}^{a(\beta_r)}\otimes v_{\lambda_i}\\ &=\sum\limits_{(\textbf{a})}\textbf{k}_\textbf{a} \Big(\lambda_i-\sum\limits^r_{i=1}a(\beta_i)\beta_i\Big) (h)x_{-\beta_1}^{a(\beta_1)}\cdot\cdot\cdot x_{-\beta_r}^{a(\beta_r)}\otimes v_{\lambda_i}. \end{align*} $

Thus, one can get $\lambda_i-r_{n-i}\alpha_n=\lambda_i-\sum\limits^r_{i=1}a(\beta_i)\beta_i$. Since $\alpha_n$ is simple root, set $\beta_j=\alpha_n$, we have $a(\alpha_n)=r_{n-i}$ and $a(\beta_i)=0, \forall \beta_i\neq\alpha_n, $ which means $f$ is the unique homomorphism between $\mbox{Hom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n), \widehat Z_\chi(\lambda_i)).$ Since $ \widehat Z_\chi(\lambda_{i+1})\cong\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n)$, we have $\hbox{dimHom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_{i+1}), \widehat Z_\chi(\lambda_i))=1.$ The lemma has been proven.

Next, consider the hom-spaces $\widehat Z_\chi(\lambda_{i}\otimes (-p(n+1)))\longrightarrow\widehat Z_\chi(\lambda_{i+1}))$.

Lemma 6.3    Assume that $0\leqslant i <n, 0< r_{n-i}\leqslant p$. Then,

$\hbox{dimHom}_{U_{\chi}({\frak g})} (\widehat Z_\chi(\lambda_i\otimes (-p(n+1))), \widehat Z_\chi(\lambda_{i+1}))\neq0 .$

Proof    If $\psi=\sum\limits_{i=1}^ni\alpha_i^\vee$, we have $\langle\alpha_n, \psi\rangle=n+1$ and $\widehat Z_\chi(\lambda_i\otimes (-p(n+1)))\cong\widehat Z_\chi(\lambda_i-p\alpha_n).$ Since $ \widehat Z_\chi(\lambda_{i+1})=\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n)$, we can get a $U_\chi({\frak g})$-module homomorphism

$ \begin{align*}f: \widehat Z_\chi(\lambda_i-p\alpha_n)&\longrightarrow\widehat Z_\chi(\lambda_i-r_{n-i}\alpha_n)\\ v_{\lambda_i-p\alpha_n}&\mapsto x_{-\alpha_n}^{p-r_{n-i}} \otimes v_{\lambda_i-r_{n-i}\alpha_n}. \end{align*} $

Furthermore, $\forall t\in T_0$, so we have

$ \begin{align*} f(t.v_{\lambda_i-p\alpha_n})& =f((\lambda_i-p\alpha_n)(t)v_{\lambda_i-p\alpha_n}) \\ &=(\lambda_i-p\alpha_n)(t)f(v_{\lambda_i-p\alpha_n}) \\ &=(\lambda_i-p\alpha_n)(t)x_{-\alpha_n}^{p-r_{n-i}}\otimes v_{\lambda_i-r_{n-i}\alpha_n}\\ &={\rm Ad}(t) (x_{-\alpha_n}^{p-r_{n-i}})\otimes(t. v_{\lambda_i-r_{n-i}\alpha_n}) \\&=t.(x_{-\alpha_n}^{p-r_{n-i}}\otimes v_{\lambda_i-r_{n-i}\alpha_n}) \\&=t.f(v_{\lambda_i-p\alpha_n}). \end{align*} $

Accordingly, we have $f$ is a $U_\chi({\frak g})$-$T_0$-module homomorphism. The lemma has been proven.

7 Hom-spaces of non-graded baby Verma modules

Let $b_i=r_{n-(i-1)}+r_{n-(i-2)}+\cdot\cdot\cdot +r_{n}$, where $\lambda_0\in C_0$ with $\lambda_0+\rho=(r_1, r_2, \cdots, r_n)$ and $r_0=p-(r_1+\cdot\cdot\cdot +r_n)$. Then, $b_0=0, b_n=p-r_0, 0\leqslant b_i\leqslant p, \forall~ i\in\{1, 2, \cdots, n\}$.

Because $\sigma=s_1s_2\cdot\cdot\cdot s_n$, we have $\sigma^{-j}(n+1)=n+1-j, \forall~0\leqslant j\leqslant n.$

Lemma 7.1    Assume that $0\leqslant b_i <p$. Then,

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_0), Z_\chi(\lambda_{i}))\neq0.$

Proof    When $b_i=0, i\leqslant n$, one can get $r_{n-(i-1)}=r_{n-(i-2)}=\cdot\cdot\cdot =r_{n}=0.$ $\forall ~j<i\leqslant n$ and we have $s_n.\lambda_j=\lambda_j.$ Then $\lambda_{j+1}=\sigma.\lambda_{j}=(s_1s_2\cdot\cdot\cdot s_n).\lambda_{j}\in W_I.\lambda_{j}.$ By Lemma 4.1, we have $ Z_\chi(\lambda_{j+1})\cong Z_\chi(\lambda_j)$. Furthermore, we get

$Z_\chi(\lambda_{0})\cong Z_\chi(\lambda_{1})\cong\cdot\cdot\cdot \cong Z_\chi(\lambda_i).$

When $0<b_i<p, i\leqslant n$, let $\lambda_i'=s_{\epsilon_{n-(i-1)}-\epsilon_n}.\lambda_0, \lambda_i''=s_n.\lambda_i'.$ Since $s_{\epsilon_{n-(i-1)}-\epsilon_n}\in {\rm Stab}_W(n+1)=W_I$, we have $\lambda_i'\in W_I.\lambda_0.$ Additionally, $Z_\chi(\lambda_i')\cong Z_\chi(\lambda_{0}).$

Because $(s_ns_{\epsilon_{n-(i-1)}-\epsilon_n})^{-1}(n+1)=n+1-i$, we get $s_ns_{\epsilon_{n-(i-1)}-\epsilon_n}\in W_I\sigma^{i}.$ Furthermore, $Z_\chi(\lambda_i'')\cong Z_\chi(\lambda_i).$

Hence, we have $\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_0), Z_\chi(\lambda_{i}))\cong\hbox{Hom}_{U_{\chi}({\frak g})}(Z_\chi(\lambda_i'), Z_\chi(\lambda_i''))\neq0 .$

By calculation, $b_i=\langle \lambda_i'+\rho, \alpha_n^\vee\rangle$, then $\lambda_i''=s_n.\lambda_i'=\lambda_i'-b_i\alpha_n.$ Construct a nonzero $U_\chi({\frak g})$-module homomorphism

$ \begin{align*} f: Z_\chi(\lambda_i')&\longrightarrow Z_\chi(\lambda_i'')\\ v_{\lambda_i'}&\mapsto x_{-\alpha_n}^{p-b_{i}}\otimes v_{\lambda_i'-b_{i}\alpha_n}, \end{align*} $

which means $\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_0), Z_\chi(\lambda_{i}))\neq0 .$ The lemma has been proven.

For another scenario, we have

Lemma 7.2    Assume that $0\leqslant b_i <p$. Then,

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_i), Z_\chi(\lambda_{0}))\neq0 .$

Proof    Because $s_n.\lambda_j=\lambda_j-r_{n-j}\alpha_n$, set

$ \begin{align*}\phi_j: Z_\chi(\lambda_j-r_{n-j}\alpha_n)&\longrightarrow Z_\chi(\lambda_j)\\ v_{\lambda_j-r_{n-j}\alpha_n}&\mapsto x_{-\alpha_n}^{r_{n-j}}\otimes v_{\lambda_j}. \end{align*} $

By Lemma 4.1, we have $ Z_\chi(\lambda_{j+1})=Z_\chi(\sigma.\lambda_{j})\cong Z_\chi(s_n.\lambda_j)= Z_\chi(\lambda_j-r_{n-j}\alpha_n)$. Let

$ \phi_j': Z_\chi(\lambda_{j+1})\longrightarrow Z_\chi(\lambda_j-r_{n-j}\alpha_n)\longrightarrow Z_\chi(\lambda_j). $

Then we get $\phi_0'\circ\phi_1'\circ\cdot\cdot\cdot \circ\phi_{i-1}'$ is the $U_\chi({\frak g})$-module homomorphism from $Z_\chi(\lambda_{i})$ to $Z_\chi(\lambda_{0})$. Since $x_{-\alpha_n}^{r_{n-(i-1)}+r_{n-(i-2)}+\cdot\cdot\cdot +r_{n}}\otimes v_{\lambda_0}=x_{-\alpha_n}^{b_{i}}\otimes v_{\lambda_0}\in \hbox{Im}\phi_0'\circ\phi_1'\circ\cdot\cdot\cdot \circ\phi_{i-1}'$, we have $\phi_0'\circ\phi_1'\circ\cdot\cdot\cdot \circ\phi_{i-1}'$ as nonzero. The lemma has been proven.

For $ b_i =p$, we have

Lemma 7.3    Assume that $ b_i =p$. Then as ${U_{\chi}({\frak g})}$-modules

$ Z_\chi(\lambda_i)\cong Z_\chi(\lambda_{0}).$

Proof    When $ b_i =p$, since $0\leqslant r_1+r_2+\cdot\cdot\cdot +r_n \leqslant p$ and $r_i\geq 0, \forall i$, we have $r_0=r_1=\cdot\cdot\cdot =r_{n-i}=0.$ For $1\leqslant k\leqslant n$, we have

$ \begin{align*} \langle\lambda_k+\rho, \alpha_n^\vee\rangle= \left\{\!\! \begin{array}{ll} r_{n-k}, & \text{for}\, k\leqslant n, \\ -(r_1+r_2+\cdots+r_n), & \text{for}\, k=n . \end{array}\right. \end{align*} $

$r_{n-k}=0$ means $s_n.\lambda_k=\lambda_k.$ Thus, $\lambda_{k+1}=\sigma.\lambda_k=(s_1s_2\cdot\cdot\cdot s_n).\lambda_k\in W_I.\lambda_k.$ By Lemma 4.1, we have $ Z_\chi(\lambda_{k+1})\cong Z_\chi(\lambda_{k})$ as ${U_{\chi}({\frak g})}$-modules. If $k=i, i+1, \cdots, $ $n-1$, we get $Z_\chi(\lambda_{i})\cong Z_\chi(\lambda_{i+1})\cong \cdot\cdot\cdot \cong Z_\chi(\lambda_{n}).$

For $r_0=0$, we have $s_n.\lambda_n=\lambda_n-\langle\lambda_n+\rho, \alpha_n^\vee\rangle\alpha_n=\lambda+(r_1+r_2+\cdot\cdot\cdot +r_n)\alpha_n=\lambda_n+p\alpha_n.$ Thus $\lambda_0=\sigma^{n+1}.\lambda_0=\sigma.\lambda_n\in W_I.(\lambda_n+p\alpha_n).$ Then we have $\widehat Z_\chi(\lambda_{0})\cong \widehat Z_\chi(\lambda_{n}+p\alpha_n)$ and $ Z_\chi(\lambda_{0})\cong Z_\chi(\lambda_{n}).$ Furthermore, one can get $ Z_\chi(\lambda_{0})\cong Z_\chi(\lambda_{i}).$ The lemma has been proven.

Corollary 7.4    Assume that $ b_i =p$. Then,

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_i), Z_\chi(\lambda_{0}))\neq0, \, \hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_0), Z_\chi(\lambda_{i}))\neq0.$

By Lemmas 7.1—7.3, and Corollary 7.4, when $\lambda_0\in \overline C_0$, we have

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_i), Z_\chi(\lambda_{0}))\neq0, \, \hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_0), Z_\chi(\lambda_{i}))\neq0.$

Let $\theta_0=\lambda_k+p\varpi_k, \theta_i=\sigma^i.\theta_0$, $1\leqslant k\leqslant n$. Then $\theta_0\in \overline C_0$ and $\lambda_{k+j}=\theta_j+p(\varpi_{k+j}-\varpi_{j})$. Furthermore, we have $Z_\chi(\lambda_{k})\cong Z_\chi(\theta_{0})$, $Z_\chi(\lambda_{k+i})\cong Z_\chi(\theta_{i}).$ Accordingly,

$ \begin{align*} &\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_k), Z_\chi(\lambda_{k+i}))\cong \hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\theta_0), Z_\chi(\theta_{i})), \\ &\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_{k+i}), Z_\chi(\lambda_k))\cong \hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi\theta_i), Z_\chi(\theta_{0})). \end{align*} $

Based on above results, we get

Theorem 7.5    Assume that $\lambda_0\in \overline C_0$. For any $i, j\in\{0, 1, 2, \cdots, n\}$, we have

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\lambda_i), Z_\chi(\lambda_{j}))\neq0.$

Because $Z_\chi(\lambda_i)$ is indecomposable and $\lambda_i\in W.\lambda_j, \forall ~i, j\in\{0, 1, 2, \cdots, n\}$, we get all $Z_\chi(\lambda_i), i=0, 1, 2, \cdots, n$, lie in the same block.

Since $\overline C_0$ is the fundamental domain for $W_p$ acting on $X(T)$, where $\lambda_0$ moves freely in $\overline C_0$, $L_\chi(\lambda_0)$ can run all the blocks of $U_\chi({\frak g})$-modules.

Because $\lambda_i, i=0, 1, 2, \cdots, n$, are the coset elements in $W/W_I$, by Lemma 4.1, we know $\{Z_\chi(\lambda_i)\, |\, i=0, 1, 2, \cdots, n\}$ is the set of all isomorphism classes of baby Verma modules in the block which include $L_\chi(\lambda_0)$. According to Theorem 7.5, we get

Theorem 7.6    For any $\nu, \eta\in \Lambda_0$,

$\hbox{Hom}_{U_{\chi}({\frak g})} ( Z_\chi(\nu), Z_\chi(\eta))\neq0\Longleftrightarrow \nu\in W.\eta.$