2.1. Basic setup

Let G be a connected semisimple non-compact Lie group with finite center. We fix once and for all a maximal compact subgroup K in G. Assume that G has a compact Cartan subgroup $T \subset K$ ; equivalently, $\text {rank} (G) = \text {rank} (K)$ . Let $\mathfrak {g}_0, \mathfrak {k}_0, \mathfrak {t}_0$ be the Lie algebras of $G, K, T$ , respectively, and $\mathfrak {g}, \mathfrak {k}, \mathfrak {t}$ be their complexifications, respectively. Let $\Phi = \Phi (\mathfrak {g}, \mathfrak {t})$ and $\Phi _c= \Phi ( \mathfrak {k} , \mathfrak {t})$ be the root systems of $(G, T)$ and $(K, T)$ , respectively, such that $\Phi _c \subset \Phi $ , and let $W_c \subset W$ be their corresponding Weyl groups, respectively. The set $\Phi _c$ is called the set of compact roots, and its complement $\Phi _n$ in $\Phi $ is called the set of non-compact roots. Let $\mathfrak {p}_0$ be the orthogonal complement of $\mathfrak {k}_0$ in $\mathfrak {g}_0$ with respect to the killing form. Denote by $\mathfrak {g}_{\alpha }$ the root space of the root $\alpha \in \Phi $ . Then $\alpha \in \Phi _c$ if and only if $\mathfrak {g}_{\alpha } \subset \mathfrak {k}$ , and $\alpha \in \Phi _n$ if and only if $\mathfrak {g}_{\alpha } \subset \mathfrak {p}$ . Moreover, $\text {dim}\ \mathfrak {p}$ is even. We fix an arbitrary positive system $\Phi ^+$ in $\Phi $ with $\Phi _c^+ = \Phi ^+ \cap \Phi _c$ , $ \Phi _n^+ = \Phi _n \cap \Phi ^+$ , $\delta = \frac {1}{2} \sum \limits _{\alpha \in \Phi ^+} \alpha $ , $\delta _c = \frac {1}{2} \sum \limits _{\alpha \in \Phi _c^+} \alpha $ , and $ \delta _n = \frac {1}{2} \sum \limits _{\alpha \in \Phi _n^+} \alpha = \delta - \delta _c$ .

Let $\mathcal {F}$ be the set of all linear forms $\lambda \in \mathfrak {t}^*$ such that $\dfrac {2 \langle \lambda - \delta , \alpha \rangle }{\langle \alpha , \alpha \rangle }$ is an integer. Such a linear form is called an integral linear form.

We define the set $\mathcal {D}(G) : = \{ \lambda \in \mathcal {F} : \langle \lambda , \alpha \rangle \neq 0\ \text {for all}\ \alpha \in \Phi ;\ \text {and}\ \langle \lambda , \alpha \rangle> 0\ \text {for all}\ \alpha \in \Phi _c^+ \}$ . An integral linear form $\lambda $ satisfying $\langle \lambda , \alpha \rangle \neq 0$ for all $\alpha \in \Phi $ is called a regular integral linear form. Note that in the definition of $\mathcal {D}(G)$ , the condition $\langle \lambda , \alpha \rangle> 0\ \text {for all}\ \alpha \in \Phi _c^+$ ensures that no two $\lambda _1, \lambda _2 \in \mathcal {D}(G)$ are $W_c$ -conjugates of each other. The following celebrated theorem of Harish-Chandra says that the set $\mathcal {D}(G)$ completely parameterizes the set of all equivalence classes of discrete series representations.

Theorem 2.1 [Reference Harish-Chandra4].

Let $\lambda \in \mathcal {D}(G)$ . Then there is a discrete series representation $\pi _{\lambda }$ of G whose global character $\Theta _{\pi _{\lambda }}$ , as a locally $L^1$ -function on G, is given on T by

$$ \begin{align*}\Theta_{\pi_{\lambda}}( e^X ) = \frac {(-1)^m \prod\limits_{\alpha \in \Phi^+} \langle \lambda , \alpha \rangle \sum\limits_{w \in W_c} \mathrm{ det~}(w)\ e^{w \lambda (X)}}{\prod\limits_{\alpha \in \Phi^+} ( e^{\frac{\alpha(X)}{2}} - e ^{- \frac{\alpha (X)}{2}})}\end{align*} $$

for $X \in \mathfrak {t}$ , and $\text {dim}\ G/K= 2m$ . Every discrete series representation is unitarily equivalent to $\pi _{\lambda }$ for some $\lambda \in \mathcal {D}(G)$ .

If an integral linear form $\lambda $ is regular, then it gives rise to a positive system $P^{\lambda } {\kern-1pt}={\kern-1pt}\{ \alpha {\kern-1pt}\in{\kern-1pt} \Phi : \langle \lambda , \alpha \rangle {\kern-1pt}>{\kern-1pt} 0 \}$ . We denote $\delta ^{\lambda } =\frac {1}{2} \sum \limits _{\alpha \in P^{\lambda }} \alpha $ .

Consider $\Gamma $ to be a uniform lattice in G. We have a right regular representation $R_{\Gamma }$ of G on $L^2(\Gamma \backslash G)$ . It decomposes as a Hilbert direct sum of unitary irreducible representations $\pi $ of G with finite multiplicity:

$$ \begin{align*}L^2(\Gamma \backslash G) \cong \widehat{\bigoplus_{\pi \in \widehat{G}}} m(\pi, \Gamma) \pi.\end{align*} $$

Here, $m(\pi , \Gamma )$ is the multiplicity of $\pi $ in $(R_\Gamma , L^2(\Gamma \backslash G))$ ; in other words, $m(\pi , \Gamma )=\text {dim}\ \mathrm { Hom}_G(\pi , R_{\Gamma }) $ . The multiplicity of a discrete series representation $\pi _{\lambda }$ with the Harish-Chandra parameter $\lambda $ is denoted by $m(\pi _{\lambda } , \Gamma )$ .

2.2. Lefchetz number and discrete series multiplicities

In this section, we define the Lefschetz number and describe the relation between the Lefschetz number and the discrete series multiplicities. We need to introduce the notion of an invariant Dirac operator on $G/K$ . The reader is referred to Chapter 6 of [Reference Williams17] for a detailed description.

We have $\mathfrak {p}= \sum \limits _{\alpha \in \Phi _n} \mathfrak {g}_{\alpha }$ . We choose a basis $\{X_i\}$ and $\{Y_j\}$ of $\mathfrak {k}$ and $\mathfrak {p}$ , respectively, such that with respect to a bilinear form B on $\mathfrak {g}$ induced from the Killing form, we have $B(X_k, X_l)= - \delta _{kl}$ and $B(Y_m, Y_n)= \delta _{mn}$ . Let $C(\mathfrak {p})$ be the Clifford algebra of $\mathfrak {p}$ with respect to the non-degenerate bilinear form B. Since $\text {dim}\ \mathfrak {p}$ is even, there is an algebra isomorphism $\epsilon $ between $C(\mathfrak {p})$ and the matrix algebra $\mathrm {End}(S)$ for some vector space S with dimension $2^{(\frac {\text {dim}\ \mathfrak {p}}{2})}$ . The restriction of $\epsilon $ to the spin group $\text {Spin}(\mathfrak {p}_0) \subset C(\mathfrak {p})$ gives a spin representation $\sigma := \epsilon |_{\text {Spin}(\mathfrak {p}_0)}$ on S. The spin representation $\sigma $ turns out to be the direct sum of two ( $\tfrac {1}{2}$ spin) irreducible representations $\sigma ^{\pm }$ on $S^{\pm }$ , and $S = S^+ \bigoplus S^-$ . Let us denote the Casimir element by $\Omega $ , which is given by $ \Omega = - \sum \limits _i X_i^2 + \sum \limits _j Y_j^2$ , in the center of the universal enveloping algebra $\mathfrak {U(g)}$ of $\mathfrak {g}$ .

Now fix a $\lambda \in \mathcal {D}(G)$ . It can be easily seen that $\lambda - \delta _c$ is $\Phi ^+_c$ -dominant. Therefore, there exists a finite-dimensional irreducible representation $\tau _{\lambda -\delta _c}$ of $\mathfrak {k}$ on a finite-dimensional vector space $V_{\lambda - \delta _c}$ with highest weight $\lambda - \delta _c$ . Let us denote $\chi _{\lambda }^{\pm } := \sigma ^{\pm } \otimes \tau _{\lambda - \delta _c}$ to be a representation of $\mathfrak {k}$ (note that $\mathfrak {k}$ is also the complexified Lie algebra of the spin group $\text {Spin}\ \mathfrak {p}_0$ ). This may not integrate to a representation of K in general. However, under the assumption that $\lambda + \delta _n^{(\lambda )}$ exponentiates to a character of T, $\chi _{\lambda }^{\pm }$ does integrate to a representation of K. We consider the smooth homogeneous vector bundle $E_{\lambda }^{\pm } := G \times _K \chi _{\lambda }^{\pm }$ on $G/K$ . Let $\Gamma ^{\infty } E^{\pm }_{\lambda }$ be the space of smooth sections of a homogeneous vector bundle $E^{\pm }_{\lambda }$ . Then there exist first-order G-invariant elliptic differential operators $D_{\lambda }^{\pm } : \Gamma ^{\infty } E_{\lambda }^{\pm } \longrightarrow \Gamma ^{\infty } E_{\lambda }^{\mp }$ . These are Dirac operators twisted by $\tau _{\lambda - \delta _c}$ . With respect to an appropriate metric, $D_{\lambda }^+$ and $D_{\lambda }^-$ are adjoints of each other, and their composition $D_{\lambda }^{\pm } D_{\lambda }^{\mp }$ has the following simple description due to Parthasarathy.

Finally, we define $H^{\pm }(E_{\lambda } , \Gamma )$ to be the space of all $\Gamma $ -invariant sections $s \in \Gamma ^{\infty } E_{\lambda }^{\pm }$ such that $D_{\lambda }^{\pm } s = 0 $ . Now we are ready to define the Lefschetz number and state the theorem that relates it with the discrete series multiplicities.

Theorem 2.7. [Reference Williams15, Theorem 3.3]

Suppose that $\lambda \in \mathcal {D}^{\star }(G)$ , and $\pi _{\lambda } \in \widehat {G}$ is the associated discrete series representation of G. Let $w_0 \in W$ be the unique Weyl group element such that $w_0 \Phi ^+ = P^{\lambda }$ , and let $l(w_0) = | w_0 (- \Phi ^+) \cap \Phi ^+ |$ be the length of $w_0$ . Then for any uniform lattice $\Gamma $ in G, the multiplicity $m(\pi _{\lambda }, \Gamma )$ of $\pi _{\lambda }$ in $L^2(\Gamma \backslash G)$ is given by

$$ \begin{align*}m(\pi_{\lambda}, \Gamma) = (-1)^{l(w_0)} \{ \text{dim} H^+(E_{\lambda} , \Gamma) - \text{dim} H^-(E_{\lambda} , \Gamma) \} = (-1)^{l(w_0)} \chi(\lambda , \Gamma).\end{align*} $$

2.3. The results of Hotta–Parthasarathy on Lefschetz number

In this section, we review a couple of results of Hotta and Parthasarathy on the Lefschetz number $\chi (\lambda , \Gamma )$ from Section 2 of [Reference Hotta and Parthasarathy5]. For any $y \in T$ , let $G_y^0$ be the identity component of the centralizer $G_y$ of y in G. Let $W_y$ be the Weyl group for the root system $\Phi (G_y^0, T)$ . Let $K_y$ be the centralizer of y in K and $d = \text {dim}\ {G/K}, d_{y}=\text {dim}\ G_y / K_y$ . We use the convention that $e^{\alpha }$ is the character of T corresponding to an element $\alpha \in \mathfrak {t}^*$ . Let $\Phi _y = \{ \alpha \in \Phi : e^{\alpha }(y)=1\}$ . Then $\Phi _y^+=\Phi ^+ \cap \Phi _y$ is a positive system of $\Phi (G_y^0, T)$ .

For any $\lambda \in \mathcal {D}(G)$ , we define the function $\Psi _{\lambda }^0$ on T as follows:

$$ \begin{align*}\Psi_{\lambda}^0(y)= \dfrac{\sum\limits_{w \in W_c/W_y} (-1)^{l(w)} e^{w \lambda -\delta} (y) \prod\limits_{\alpha \in \Phi_y^+} \langle w \lambda , \alpha \rangle}{\prod\limits_{\alpha \in \Phi^+ \setminus \Phi_y^+} (1 - e^{-\alpha}(y))}.\end{align*} $$

Here, $\delta =\frac {1}{2}\sum \limits _{\alpha \in \Phi ^+} \alpha $ , and $W_y$ is considered as a subgroup of $W_c$ . The function $\Psi _{\lambda }^0$ is $W_c$ -invariant in $\lambda $ , and it extends to an $\text {Ad}(G)$ -invariant function on the set $G_e$ of elliptic elements of G, that is, the set of elements of G conjugate to some element of K.

Finally, for any $\lambda \in \mathcal {D}(G)$ , we define the function $\Psi _{\lambda }(y)$ in y as follows:

$$ \begin{align*}\Psi_{\lambda}(y)=\dfrac{ (-1)^{\frac{d + d_y}{2}} \prod\limits_{\alpha \in \Phi_y^+} (\langle \delta_y, \alpha \rangle)^{-1}\ \Psi_{\lambda}^0(y)}{|W_y| [G_y : G_y^0] }.\end{align*} $$

For any element $y \in \Gamma $ , we let $\Gamma _y$ denote the centralizer of y. Then we have the following result of Hotta and Parthasarathy.

Theorem 2.9. [Reference Hotta and Parthasarathy5, Theorem 3]

Let $\Gamma $ be a uniform lattice in G. For an arbitrary $\lambda \in \mathcal {D}(G)$ , we have

$$ \begin{align*}\chi(\lambda, \Gamma)= (-1)^{\ell(w_o)}\sum\limits_{[y]} \text{vol} (\Gamma_y \backslash G_y) \Psi_{\lambda}(y),\end{align*} $$

where the sum runs over the (finite) set of $\Gamma $ -conjugacy classes of elliptic elements in $\Gamma $ and $w_0$ is the unique element in W such that $w_0 \Phi ^+ = P^{\lambda }$ .

Combining Theorems 2.7 and 2.9, we deduce the following.

Theorem 2.11. Let $\lambda \in \mathcal {D}^{\star }(G)$ and $w_0 \in W$ such that $w_0 \Phi ^+=P^{(\lambda )}$ . Then

$$ \begin{align*}m(\pi_{\lambda}, \Gamma) =\sum\limits_{[y]} \text{vol}(\Gamma_y \backslash G_y) \Psi_{\lambda}(y),\end{align*} $$

where the sum runs over the set of $\Gamma $ -conjugacy classes of elliptic elements in $\Gamma $ .

2.4. String of regular integral linear forms

In this section, we introduce the notion of a string of regular integral linear forms in $\mathcal {D}(G)$ and $\mathcal {D}^{\star }(G)$ .

For $\lambda _1 , \lambda _2 \in \mathcal {D}(G)$ , let $P^{\lambda _1}$ and $P^{\lambda _2}$ be the corresponding positive root systems for the regular linear forms $\lambda _1$ and $\lambda _2$ , respectively. If $P^{\lambda _1} \neq P^{\lambda _2}$ , then there exists $\sigma \in W$ such that $\sigma P^{\lambda _1} =P^{\lambda _2}$ . On the other hand, $\sigma P^{\lambda _1} = P^{\sigma \lambda _1}$ . Therefore, without loss of generality, we can consider $\lambda _1, \lambda _2 \in \mathcal {D}(G)$ such that $P^{\lambda _1} = P^{\lambda _2}$ .

Proof. For any $k \in {\mathbb N}$ , it is easy to see that $k \lambda \in \mathcal {D}(G)$ and $P^{\lambda } = P^{k \lambda }$ , that is, $\lambda $ and $k \lambda $ determine the same positive chamber. We have

(2.1) $$ \begin{align} \begin{aligned} \langle k \lambda - \delta^{k \lambda}, \alpha \rangle & = \langle (k-1) \lambda , \alpha \rangle + \langle \lambda - \delta^{\lambda}, \alpha \rangle \\ & = (k-1) \langle \lambda , \alpha \rangle + \langle \lambda - \delta^{\lambda}, \alpha \rangle.\\ \end{aligned} \end{align} $$

Now for any $\alpha \in P^{\lambda } \cap \Phi _n$ , $\langle \lambda , \alpha \rangle> 0$ and by the hypothesis, $\langle \lambda - \delta ^{\lambda }, \alpha \rangle> 0$ . Hence, $k \lambda \in \mathcal {D}^{\star }(G)$ .

Before describing the next lemma, we record a useful observation. If $\lambda _1, \lambda _2 \in \mathcal {D}(G)$ , then one can easily check that $P^{(\lambda _1)} \cap P^{(\lambda _2)} \subset P^{(\lambda _1 + \lambda _2)} \subset P^{(\lambda _1)} \cup P^{(\lambda _2)}$ .

Proof. Indeed, $\lambda _1 + k \lambda _2 \in \mathcal {D}(G)$ for any $k \in {\mathbb N}$ and $P^{(\lambda _1)} \cap P^{(k \lambda _2)} \subset P^{(\lambda _1 + k \lambda _2)} \subset P^{(\lambda _1)} \cup P^{(k \lambda _2)}$ . Therefore, all integral linear forms of the string determine the same positive system as the one determined by the base and the direction of the string. Without loss of generality, we assume that $\lambda _2 \in \mathcal {D}^{\star }(G)$ . Let us write $P_n^{\lambda } =P^{\lambda } \cap \Phi _n$ . Then for any $\alpha \in P_n^{( \lambda _1 + k \lambda _2)} = P_n^{(k \lambda _2)} = P_n^{( \lambda _1)} \subset P^{(\lambda _1)}$ , we have

$$ \begin{align*}\langle \lambda_1 + k \lambda_2 - \delta^{(\lambda_1 + k \lambda_2)}, \alpha \rangle = \langle \lambda_1 , \alpha \rangle + \langle k \lambda_2 - \delta^{(k \lambda_2)} , \alpha \rangle> 0.\end{align*} $$

Therefore, every member of the string $\mathcal {S}(\lambda _1, \lambda _2)$ belongs to $\mathcal {D}^{\star }(G)$ .

4.1. Proof of Theorem 1.2

For simplicity, we write $N:=N_{\Gamma }$ . We have the string $\mathcal {S}(\lambda _1, \lambda _2)$ with base $\lambda _1$ , direction $\lambda _2$ and ${P^{\lambda _1}=P^{\lambda _2}}$ ; and by Proposition 3.1, its generating function $F_{\lambda _1, \lambda _2, \Gamma }$ is of the form

$$ \begin{align*}F_{\lambda_1, \lambda_2, \Gamma } (z) = \frac{p(z)}{(1-z^{N})^{|\Phi^+|+1}},\end{align*} $$

with deg $p(z) < N(|\Phi ^+|+1)$ . Let us write $p(z) = \sum \limits _{k=0}^{N(|\Phi ^+|+1)-1} b_k z^k$ for some $b_k \in {\mathbb C}$ . We claim that for any $0 \leq j \leq N-1$ and $m \geq 0$ ,

$$ \begin{align*}m(\pi_{\lambda_{mN+j}},\Gamma) = \sum\limits_{h=0}^{|\Phi^+|} b_{hN+j} \left( \begin{matrix} m-h+|\Phi^+| \\ |\Phi^+| \end{matrix} \right),\end{align*} $$

where $ \left ( \begin {matrix} m-h+|\Phi ^+| \\ |\Phi ^+| \end {matrix} \right ) =0\ \text {when}\ m<h$ . Therefore, the sum essentially runs over $0 \leq h \leq \text {min}(|\Phi ^+|,m)$ . We use the following binomial expansion:

$$ \begin{align*}\frac{1}{(1-y)^{j+1}} = \sum_{k \geq 0} \left( \begin{matrix} k+j \\ k \end{matrix} \right) y^k \hspace{10pt} \text{for any}\ j \in {\mathbb N}.\end{align*} $$

Then, we have

$$ \begin{align*} F_{\lambda_1,\lambda_2,\Gamma} (z) & = p(z) (1-z^{N})^{-(|\Phi^+|+1)} \\& = \left(\sum_{j=0}^{N-1} \sum_{h=0}^{|\Phi^+|} b_{hN+j} z^{hN+j} \right) \left( \sum_{k \geq0} \left( \begin{matrix} k+ |\Phi^+| \\ |\Phi^+| \end{matrix} \right) z^{kN} \right) \\& = \sum_{j=0}^{N-1} \sum_{k\geq0} \sum_{h=0}^{|\Phi^+|} b_{hN+j} \left( \begin{matrix} k+ |\Phi^+| \\ |\Phi^+| \end{matrix} \right) z^{(k+h)N +j} \\& = \sum_{j=0}^{N-1} \sum_{m\geq0} \left( \sum_{h=0}^{|\Phi^+|} b_{hN+j} \left( \begin{matrix} m-h + |\Phi^+| \\ |\Phi^+| \end{matrix} \right) \right) z^{mN+j}. \end{align*} $$

Let us fix a j with $0 \leq j \leq N-1$ . Since $|\mathcal {A} \cap (j+N {\mathbb Z})| \geq |\Phi ^+|+1$ , there are $m_0,\dots ,m_{|\Phi ^+|} \in {\mathbb Z}^+$ such that $m_i N+j \in \mathcal {A}$ for all $0 \leq i \leq |\Phi ^+|$ .

Therefore,

$$ \begin{align*}m(\pi_{\lambda_{m_i N +j}},\Gamma) = \sum_{h=0}^{|\Phi^+|} b_{hN+j} \left( \begin{matrix} m_i - h + |\Phi^+| \\ |\Phi^+| \end{matrix} \right)\hspace{15pt} \text{with }\ 0 \leq i \leq |\Phi^+|\end{align*} $$

is a system of $|\Phi ^+| +1$ linear equations in $|\Phi ^+| +1$ variables $\left \{ b_{hN+j} \right \}_{h=0}^{|\Phi ^+|}$ . It is not difficult to check that the square matrix

$$ \begin{align*}\Biggl \{ { m_i - h + |\Phi^+ | \choose |\Phi^+| }\Biggr \} _{0 ~\leq~ h,~ i ~\leq ~|\Phi^+|}\end{align*} $$

of order $|\Phi ^+| + 1$ is non-singular. Hence, the system has a unique solution. Consequently, for any $0 \leq h \leq |\Phi ^+|$ , $b_{hN+j}=\sum \limits _i n(hN+j, m_iN+j)\ m(\pi _{\lambda _{m_i N +j}},\Gamma )$ for some integers $n(hN+j, m_iN+j)$ independent of $\Gamma $ . In other words, the solution $\left \{ b_{hN+j} \right \}_{h=0}^{|\Phi ^+|}$ can be expressed linearly in terms of $\{ m(\pi _{\lambda _{m_i N +j}},\Gamma )$ : $m_iN+j \in \mathcal {A}\} $ .

Since the finite set $\mathcal {A}$ satisfies the condition $|\mathcal {A} \cap (j+N {\mathbb Z})| \geq |\Phi ^+|+1$ for all $j \in \left \{ 0,1, \dots ,N-1 \right \}$ , we have $ b_{\ell } = \sum \limits _{k \in \mathcal {A}} n(\ell , k)\ m(\pi _{\lambda _k},\Gamma )$ for all $0 \leq \ell < N(|\Phi ^+|+1)$ . This implies that the polynomial $p(z)$ and hence the generating function $F_{\lambda _1, \lambda _2, \Gamma }$ is determined by the same. Therefore, for any $\ell \geq 0$ , the discrete series multiplicity $m(\pi _{\lambda _{\ell }},\Gamma )=\sum \limits _{k \in \mathcal {A}} n(\ell , k)\ m(\pi _{\lambda _k},\Gamma )$ for some integers $n(\ell , k)$ independent of $\Gamma $ .

4.2. Proof of Corollary 1.5

Let $\mathcal {A}= \left \{k \in {\mathbb N} \cup \{0\} : m(\pi _{\lambda _k},\Gamma _1)=m(\pi _{\lambda _k},\Gamma _2) \right \}$ . We need to show that $\mathcal {A}$ satisfies

$$ \begin{align*}|\mathcal{A} \cap (N {\mathbb Z} +j)| \geq |\Phi^+|+1 \hspace{10pt} \text{for all} \hspace{5pt} 0 \leq j \leq N-1.\end{align*} $$

By the hypothesis, we have

$$ \begin{align*}\liminf\limits_{t \rightarrow \infty} \frac{| \left\{ 0 \leq k \leq t : k \in \mathcal{A} \right\}|}{t}> \frac{N-1}{N}.\end{align*} $$

This means that for every $\epsilon>0$ , there is $r_0>0$ such that

$$ \begin{align*}\frac{|\left\{ k \in \mathcal{A} : k<rN(|\Phi^+|+1) \right\}|}{rN(|\Phi^+|+1)}>\frac{N-1+\epsilon}{N} \hspace{10pt} \text{for all}~r \geq r_0.\end{align*} $$

Equivalently,

$$ \begin{align*}| \left\{ k \in \mathcal{A} : k<rN(|\Phi^+|+1) \right\}|>r(N-1+\epsilon) (|\Phi^+|+1).\end{align*} $$

Let $r \geq r_0$ . Fix a $j_0$ such that $0 \leq j_0 \leq N-1$ . Then the cardinality

$$ \begin{align*} \begin{aligned} | \left\{ k \in \mathcal{A} : k<rN(|\Phi^+|+1) \right\}| &= \sum_{j=0}^{N-1} |\left\{ k \in \mathcal{A} : k<rN(|\Phi^+|+1) \right\} \cap (j+N {\mathbb Z})|\\ &\leq |\mathcal{A} \cap (j_0 + N{\mathbb Z})| + (N-1)r (|\Phi^+|+1). \end{aligned} \end{align*} $$

In other words,

$$ \begin{align*}|\mathcal{A} \cap (j_0 + N{\mathbb Z})|> r \epsilon (|\Phi^+|+1).\end{align*} $$

This holds for all $r \geq r_0$ . In particular, for $r \geq \frac {1}{\epsilon }$ , we have $|\mathcal {A} \cap (j_0+ N{\mathbb Z})| \geq (|\Phi ^+|+1)$ . This holds for all $j_0 \in \left \{ 0,1, \dots ,N-1 \right \}$ .

4.3. Exhaustion of Harish-Chandra parameters in $\mathcal {D}^{\star }(G)$

In this section, we describe the notion of a multi-directed string of Harish-Chandra parameters and prove analogous result to Theorem 1.2 in this generality. Let us denote the Cartesian product ${\mathbb N} \times {\mathbb N} \times \dots \times {\mathbb N}$ of n copies of ${\mathbb N}$ by ${\mathbb N}^n$ .

It is not difficult to see the following generalization of Lemma 2.16.

Now we can generalize Theorem 1.2 for a multi-directed string. First, we need a few notations: Let $p_i : {\mathbb N}^n \longrightarrow {\mathbb N}$ be the projection onto ith component of an n-tuple of natural numbers. For a subset $\boldsymbol {\mathcal {A}}$ of ${\mathbb N}^n$ , let $\mathcal {A}_i$ denote the image of $\boldsymbol {\mathcal {A}}$ under $p_i$ for $1 \leq i \leq n$ .

It is tempting to ask whether there exists a finite set $\{\lambda _1, \lambda _2, \dots ,\lambda _n\} \subset \mathcal {D}(G)$ , with the property $P^{\lambda _r}=P^{\lambda _s}$ for all $1 \leq r,s \leq n$ and at least one of the $\lambda _r$ ’s belongs to $\mathcal {D}^{\star }(G)$ , so that the set equality $\{ \mu \in \mathcal {D}^{\star }(G): P^{\mu }=P^{\lambda _r} \}=\mathcal {S}(\lambda _1, \lambda _2, \dots ,\lambda _n)$ holds. The inclusion $\mathcal {S}(\lambda _1, \lambda _2, \dots , \lambda _n) \subset \{ \mu \in \mathcal {D}^{\star }(G) : P^{\mu }=P^{\lambda _r} \}$ is obvious, whereas the reverse containment is not true by the following proposition.