Global properties of a Hecke ring associated with the Heisenberg Lie algebra

This study concerns Hecke rings introduced by Shimura [15]. A classical study of Hecke rings is the work by Hecke [6] and Tamagawa [18] on the Hecke rings associated with the general linear groups. They showed that these Hecke rings are commutative polynomial rings. Furthermore, they defined formal power series with coefficients in these Hecke rings, and showed their rationality. The results of this work are summarized in [17, Chapter 3], where formal Dirichlet series with coefficients in these Hecke rings were further introduced. Andrianov [1], Hina–Sugano [7], Satake [12], and Shimura [16] studied Hecke rings associated with classical groups, wherein they further developed the work of Hecke [6] and Tamagawa [18]. In addition, other studies were conducted on the Hecke rings associated with Jacobi and Chevalley groups by Dulinsky [4] and Iwahori-Matsumoto [11], respectively.

As mentioned above, various studies have been carried out on Hecke rings. However, the class of Hecke rings defined by Shimura is vast, and only a small part of it has been studied to date.

From now on, an algebra implies an abelian group with a bi-additive product (e.g., an associative algebra, a Lie algebra). Let $L$ be an algebra that is free of finite rank as an abelian group. Our previous work [9] introduced the Hecke rings $R_{L}$ and $\widehat{R}_{L}$ associated with $L$. For the definition, see Section 2. In this study, we deal with the formal Dirichlet series $D_{L}(s)$ and $\widehat{D}_{L}(s)$ with coefficients in $R_{L}$ and $\widehat{R}_{L}$, respectively, which are defined in Section 3.

The first result of this study is to show that the Euler product formula for $\widehat{D}_{L}(s)$ holds, and to give a sufficient condition for $D_{L}(s)$ to have the Euler product expansion (cf. Theorems 3.1 and 3.2).

If $L=\mathbb{Z}^{r}$ is the free abelian Lie algebra of rank $r$, the Hecke ring $R_{\mathbb{Z}^{r}}$ and $\widehat{R}_{\mathbb{Z}^{r}}$ coincide with those treated by Hecke [6] and Tamagawa [18]. Further, the formal Dirichlet series $D_{\mathbb{Z}^{r}}(s)$ and $\widehat{D}_{\mathbb{Z}^{r}}(s)$ equal those treated in [17, Chapter 3]. Thus, it can be said that our study generalizes their study. We discuss them in Section 4.

Denote by $\mathcal{H}$ the Heisenberg Lie algebra, that is, the free nilpotent Lie algebra of class $2$ on two generators. The second result of this study is the establishment of identities for $D_{\mathcal{H}}(s)$ and $\widehat{D}_{\mathcal{H}}(s)$, which is the primary result of this study. Let $\widehat{\boldsymbol{\theta}}=(\widehat{\theta}_{p})_{p}$ be a family of indeterminates indexed by all prime numbers $p$. The key idea for stating our main theorem involves regarding $\widehat{R}_{\mathcal{H}}$ as a module over the polynomial ring $\widehat{R}_{\mathbb{Z}^{2}}[\widehat{\boldsymbol{\theta}}]$. The main theorem is as follows:

It is worth noting that this theorem is similar to Shimura’s Theorem 4.5 for the case $r=2$. At the conclusion of Section 5.2, we establish that Theorem 1.1 recovers Shimura’s Theorem for $r=2$ via the endomorphism $\widehat{\phi}$ introduced in Definition 5.10.

The proof is essentially done by using some results of our previous study [8] which is described in Section 5.1. There is no great difficulty in proving the claims stated in this study. Rather, it is important to note a natural generalization of series of [6, 17, 18], and a concise identity given for a formal Dirichlet series whose coefficients are not always commutative (cf. Remark 5.14).

In [8, 9] and this study, the case of the Heisenberg Lie algebra is considered as a first step. The author expects that many new Hecke rings will appear in the class of the Hecke rings of this study. Further study of these Hecke rings is now in progress by the author. In [10], the author investigated the Euler factor of $\widehat{D}_{L}(s)$ at each prime number in the case where $L$ is a higher Heisenberg Lie algebra.

Tamagawa [18], by using his Hecke theory, further investigated certain zeta functions, and proved that each of them is an entire function and has a functional equation. However, even in the Hecke rings associated with the Heisenberg Lie algebra, no analogue has been found. Further research is needed to find such applications to number theory.

It should be mentioned that our series $D_{L}(s)$ and $\widehat{D}_{L}(s)$ are related to the zeta functions of groups and rings introduced by Grunewald, Segal and Smith [5]. Let $G$ be a torsion-free finitely generated nilpotent group, and $\widehat{G}$ its profinite completion. Denote by $\mathcal{S}^{i}_{n}(G)\ (\text{resp.}\ \mathcal{S}^{\wedge}_{n}(G))$ the family of subgroups $H$ of $G$ of index $n$ such that there is an isomorphism $H\cong G$ of groups $(\text{resp.}\ \widehat{H}\cong\widehat{G}\ \text{of topological groups})$. The zeta functions $\zeta^{i}_{G}(s)$ and $\zeta^{\wedge}_{G}(s)$ of $G$ were defined in [5] as follows:

$$\zeta^{i}_{G}(s)=\sum_{n>0}\#\mathcal{S}^{i}_{n}(G)n^{-s},\quad\zeta^{\wedge}_{G}(s)=\sum_{n>0}\#\mathcal{S}^{\wedge}_{n}(G)n^{-s},$$

where $s$ is a complex variable.

As an analogue of them, one can define the zeta functions of $L$. Let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and set $\widehat{L}=L\otimes\widehat{\mathbb{Z}}$. Denote by $\mathcal{S}^{i}_{n}(L)\ (\text{resp.}\ \mathcal{S}^{\wedge}_{n}(L))$ the family of subalgebras $M$ of $L$ of index $n$ such that there is an isomorphism $M\cong L$ as algebras $(\text{resp.}\ M\otimes\widehat{\mathbb{Z}}\cong\widehat{L}\ \text{as algebras over $\widehat{\mathbb{Z}}$})$. We set $a_{n}^{i}(L)=\#\mathcal{S}^{i}_{n}(L)$ and $a_{n}^{\wedge}(L)=\#\mathcal{S}^{\wedge}_{n}(L)$ for each $n$. The zeta functions $\zeta^{i}_{L}(s)$ and $\zeta_{L}^{\wedge}(s)$ of $L$ are defined as follows:

$$\zeta^{i}_{L}(s)=\sum_{n>0}a_{n}^{i}(L)n^{-s},\quad\zeta_{L}^{\wedge}(s)=\sum_{n>0}a_{n}^{\wedge}(L)n^{-s}.$$

The zeta function $\zeta_{L}^{\wedge}(s)$ was also introduced in [5], and is called pro-isomorphic zeta function $\zeta_{L}^{\wedge}(s)$ of $L$ in [2] and [3]. Although there are few papers on $\zeta^{i}_{L}(s)$, it is a natural analogue of $\zeta^{i}_{G}(s)$. We call $\zeta^{i}_{L}(s)$ the isomorphic zeta function of $L$.

As we mention in Section 6, $\zeta^{i}_{L}(s)$ and $\zeta_{L}^{\wedge}(s)$ equal the coefficient-wise images of $D_{L}(s)$ and $\widehat{D}_{L}(s)$ under the degree maps on $R_{L}$ and $\widehat{R}_{L}$, respectively. For the definition of the degree map on a Hecke ring, see Section 2. Moreover, at the end of Section 6, we prove that our Theorem 1.1 derives the explicit formulae for $\zeta^{i}_{\mathcal{H}}(s)$ and $\zeta_{\mathcal{H}}^{\wedge}(s)$ via the degree map on $\widehat{R}_{\mathcal{H}}$ as follows:

$$\zeta^{i}_{\mathcal{H}}(s)=\zeta_{\mathcal{H}}^{\wedge}(s)=\zeta(2s-2)\zeta(2s-3),$$

where $\zeta(s)$ is the Riemann zeta function.

This identity is essentially due to Grunewald, Segal and Smith [5, Theorem 7.6]. Precisely, for the free nilpotent group $\mathfrak{F}=\mathfrak{F}_{c,g}$ of class $c$ on $g$-generators, the identity $\zeta^{i}_{\mathfrak{F}}(s)=\zeta_{\mathfrak{F}}^{\wedge}(s)$ and the explicit formulae for them were obtained. For the free nilpotent Lie algebra $\mathcal{F}=\mathcal{F}_{c,g}$ of class $c$ on $g$-generators, by an argument essentially equivalent to that of [5], Berman, Glazer, and Schein proved in [2, Theorem 5.1] that $\zeta_{\mathcal{F}}^{\wedge}(s)$ equals $\zeta^{i}_{\mathfrak{F}}(s)$ and $\zeta^{\wedge}_{\mathfrak{F}}(s)$ (cf. Theorem 7.1). In Proposition 7.4, the equality $\zeta^{\wedge}_{\mathcal{F}}(s)=\zeta^{i}_{\mathcal{F}}(s)$ is verified in a similar way as in the proof of [5, Theorem 7.6]. As a result, the equality $\zeta^{i}_{\mathcal{F}}(s)=\zeta_{\mathcal{F}}^{\wedge}(s)=\zeta_{\mathfrak{F}}^{\wedge}(s)=\zeta^{i}_{\mathfrak{F}}(s)$ holds.

Write $\mathfrak{H}$ for the Heisenberg group $\mathfrak{F}_{2,2}$, and focus on the identity $\zeta^{i}_{\mathcal{H}}(s)=\zeta_{\mathcal{H}}^{\wedge}(s)=\zeta^{\wedge}_{\mathfrak{H}}(s)=\zeta^{i}_{\mathfrak{H}}(s)$. Another generalization of the identity $\zeta_{\mathcal{H}}^{\wedge}(s)=\zeta^{\wedge}_{\mathfrak{H}}(s)$ is known. Suppose that the nilpotent class of $G$ is $2$. Define the Lie algebra $L(G)$ as $(G/Z)\oplus Z$ with the usual Lie bracket operation induced by the commutator in $G$, where $Z$ is the center of $G$. Then, we have $L(\mathfrak{H})=\mathcal{H}$, and the identity $\zeta^{\wedge}_{G}(s)=\zeta^{\wedge}_{L(G)}(s)$ is known to hold (cf. [2, Section 1.1] or [13, Section 1.2.2]). On the other hand, $\zeta^{i}_{G}(s)=\zeta_{G}^{\wedge}(s)$ does not hold in general. Indeed, Theorems 7.1 and 7.3 of [5] provide a counter-example. The equality $\zeta^{i}_{G}(s)=\zeta^{i}_{L(G)}(s)$ is proved in Corollary 7.6, and thus $\zeta_{L(G)}^{\wedge}(s)=\zeta^{i}_{L(G)}(s)$ does not always hold.

For pro-isomorphic zeta functions of Lie algebras, Berman, Glazer, and Schein [2] further investigated. The explicit formula for $\zeta_{L}^{\wedge}(s)$ was shown in [2, Section 5], specifically for $L$ belonging to a certain class of Lie algebras over the integer rings of number fields. So far, we have not found any formulae for $D_{L}(s)$ and $\widehat{D}_{L}(s)$ that recover their formulae except for this study.

The contents of this paper are organized as follows. In Section 2, we review the Hecke rings $R_{L}$ and $\widehat{R}_{L}$. In Section 3, the formal series $D_{L}(s)$ and $\widehat{D}_{L}(s)$ are introduced. In Section 4, the case of $L=\mathbb{Z}^{r}$ is considered. In Section 5, we study the series $D_{\mathcal{H}}(s)$ and $\widehat{D}_{\mathcal{H}}(s)$, and prove our main theorem. In Section 6, our series $D_{L}(s)$ and $\widehat{D}_{L}(s)$ are related to the isomorphic zeta function $\zeta^{i}_{L}(s)$ and the pro-isomorphic zeta function $\zeta_{L}^{\wedge}(s)$ of $L$, respectively. Subsequently, we prove that our main theorem also recovers the explicit formulae for $\zeta^{i}_{\mathcal{H}}(s)$ and $\zeta_{\mathcal{H}}^{\wedge}(s)$. Finally, in Section 7, we observe the isomorphic zeta functions in the cases of the free nilpotent Lie algebras and class-2 nilpotent Lie algebras.

First, we briefly recall the definition of Hecke rings and their degree maps. For more details, refer to [17, Chapter 3]. Let $G$ be a group, $\Delta$ be a submonoid of $G$, and $\Gamma$ be a subgroup of $\Delta$. We assume that the pair $(\Gamma,\Delta)$ is a double finite pair; that is, for all $A\in\Delta$, $\Gamma\backslash\Gamma A\Gamma$ and $\Gamma A\Gamma/\Gamma$ are finite sets. Then, one can define the Hecke ring $R=R(\Gamma,\Delta)$ associated with the pair $(\Gamma,\Delta)$ as follows:

• •

  The underlying abelian group is the free abelian group on the set $\Gamma\backslash\Delta/\Gamma$.

• •

  For all $A,B\in\Delta$, the product of $\Gamma A\Gamma$ and $\Gamma B\Gamma$ is defined to be

  $$\sum_{\Gamma C\Gamma\in\Gamma\backslash\Delta/\Gamma}\#\{\Gamma\beta\in\Gamma\backslash\Gamma B\Gamma\ |\ C\beta^{-1}\in\Gamma A\Gamma\}\cdot\Gamma C\Gamma.$$

For every $A\in\Delta$, write $T_{\Gamma,\Delta}(A)$ for the element $\Gamma A\Gamma$ of $R$. We define the degree map on $R$ to be the additive map $\deg\!{R}:R\to\mathbb{Z}$ such that $T_{\Gamma,\Delta}(A)^{\deg\!{R}}=\#\Gamma\backslash\Gamma A\Gamma$ for every $A\in\Delta$. Notably, it is known that $\deg\!{R}$ forms a ring homomorphism.

Let $p$ be a prime number, and let $L$ be as in Section 1. We next recall the Hecke rings associated with $L$ introduced in [9]. Fix a $\mathbb{Z}$-basis of $L$, and let $r$ be the rank of $L$. Then, ${\rm Aut}_{\mathbb{Q}}^{alg}(L\otimes\mathbb{Q})$, ${\rm Aut}_{\mathbb{Q}_{p}}^{alg}(L\otimes\mathbb{Q}_{p})$, ${\rm End}_{\mathbb{Z}}^{alg}(L)$, and ${\rm End}_{\mathbb{Z}_{p}}^{alg}(L\otimes\mathbb{Z}_{p})$ are all identified with subsets of $M_{r}(\mathbb{Q}_{p})$. In [9, Section 2], the following notation was introduced:

$$\begin{array}[]{llllll}G_{L}&=&{\rm Aut}_{\mathbb{Q}}^{alg}(L\otimes\mathbb{Q}),&G_{L_{p}}&=&{\rm Aut}_{\mathbb{Q}_{p}}^{alg}(L\otimes\mathbb{Q}_{p}),\\ \rule{0.0pt}{12.91663pt}\Delta_{L}&=&{\rm End}_{\mathbb{Z}}^{alg}(L)\cap G_{L},&\Delta_{L_{p}}&=&{\rm End}_{\mathbb{Z}_{p}}^{alg}(L\otimes\mathbb{Z}_{p})\cap G_{L_{p}},\\ \rule{0.0pt}{12.91663pt}\Gamma_{\!L}&=&{\rm Aut}_{\mathbb{Z}}^{alg}(L),&\Gamma_{\!L_{p}}&=&{\rm Aut}_{\mathbb{Z}_{p}}^{alg}(L\otimes\mathbb{Z}_{p}).\end{array}$$

The global Hecke rings $R_{L}$ and the local Hecke ring $R_{L_{p}}$ are the Hecke rings with respect to $(\Gamma_{\!L},\Delta_{L})$ and $(\Gamma_{\!L_{p}},\Delta_{L_{p}})$, respectively.

The other global Hecke ring $\widehat{R}_{L}$ was introduced in [9, Section 3]. Define the group $\widehat{G}_{L}$ to be the restricted direct product of $G_{L_{p}}$ relative to $\Gamma_{\!L_{p}}$ for all prime numbers $p$, that is, the set of elements $(\alpha_{p})_{p}$ of $\prod_{p}G_{L_{p}}$ such that $\alpha_{p}\in\Gamma_{\!L_{p}}$ for almost all $p$. The monoid $\widehat{\Delta}_{L}$ and the group $\widehat{\Gamma}_{\!L}$ denote $\widehat{G}_{L}\cap\prod_{p}\Delta_{L_{p}}$ and $\prod_{p}\Gamma_{\!L_{p}}$, respectively. Then, we write $\widehat{R}_{L}$ for the Hecke ring with respect to $(\widehat{\Gamma}_{\!L},\widehat{\Delta}_{L})$.

Section 3 of [9] described relations among these Hecke rings. The local Hecke ring $R_{L_{p}}$ is related to the global Hecke ring $\widehat{R}_{L}$ as follows:

For simplicity, we set

$$T_{L_{p}}=T_{\Gamma_{\!L_{p}},\Delta_{L_{p}}},\quad T_{L}=T_{\Gamma_{\!L},\Delta_{L}},\quad\widehat{T}_{L}=T_{\widehat{\Gamma}_{\!L},\widehat{\Delta}_{L}}.$$

Then, we have $T_{L_{p}}(\alpha)=\widehat{T}_{L}(\alpha)$ in $\widehat{R}_{L}$ for each $\alpha\in\Delta_{L_{p}}$. Let us relate the global Hecke rings $\widehat{R}_{L}$ and $R_{L}$. The map $\eta_{L}$ denotes the diagonal embedding of $\Delta_{L}$ into $\prod_{p}\Delta_{L_{p}}$. Then, we define the additive map $\eta^{*}_{L}:\widehat{R}_{L}\to R_{L}$ given by $\widehat{T}_{L}(\widehat{\alpha})\mapsto\sum_{\beta}T_{L}(\beta)$, where $\beta$ runs through a complete system of representatives of $\Gamma_{\!L}\backslash\eta_{L}^{-1}(\widehat{\Gamma}_{\!L}\widehat{\alpha}\widehat{\Gamma}_{\!L})/\Gamma_{\!L}$. Let us denote by $\eta_{*L}:\Gamma_{\!L}\backslash\Delta_{L}\to\widehat{\Gamma}_{\!L}\backslash\widehat{\Delta}_{L}$ the map induced by $\eta_{L}$. Then, the two global Hecke rings are related as follows:

From now on, we regard $\widehat{R}_{L}$ as a subring of $R_{L}$ if $\eta_{*L}$ is bijective.

In the rest of this section, we relate $\widehat{R}_{L}$ to the automorphism group of $\widehat{L}$.

Let $p$ and $L$ be as in the previous section. We set $L_{p}=L\otimes\mathbb{Z}_{p}$. In this section, the formal series $P_{L_{p}}(X)$, $D_{L}(s)$, and $\widehat{D}_{L}(s)$ are defined. Subsequently, their relationship is described. For a positive integer $n$ and a nonnegative integer $k$, we introduce the following notation:

	$\displaystyle\widehat{\mathcal{A}}_{L}(n)$	$\displaystyle=\left\{\widehat{\alpha}\in\widehat{\Delta}_{L}\ \left|\ [\widehat{L}:\widehat{L}^{\widehat{\alpha}}]=n\right.\right\},\ \mathcal{A}_{L}(n)=\left\{\alpha\in\Delta_{L}\ \left|\ [L:L^{\alpha}]=n\right.\right\},$
	$\displaystyle\mathcal{A}_{L_{p}}(p^{k})$	$\displaystyle=\left\{\alpha\in\Delta_{L_{p}}\ \left|\ [L_{p}:L_{p}^{\alpha}]=p^{k}\right.\right\},$

where $L_{p}^{\alpha}$ is the image of $L_{p}$ under the endomorphism $\alpha$. Additionally, $\widehat{L}^{\widehat{\alpha}}$ and $L^{\alpha}$ are defined in a similar manner. Note that each element of $\widehat{\Delta}_{L}$ is regarded as an element of ${\rm End}_{\widehat{\mathbb{Z}}}^{alg}(\widehat{L})$ by Proposition 2.3.

Now, the formal power series $P_{L_{p}}(X)$ is introduced. We define

$$T_{L_{p}}(p^{k})=\sum_{\alpha}T_{L_{p}}(\alpha),\\$$

where $\alpha$ runs through a complete system of representatives of $\Gamma_{\!L_{p}}\backslash\mathcal{A}_{L_{p}}(p^{k})/\Gamma_{\!L_{p}}$. The formal power series $P_{L_{p}}(X)$ is defined as the generating function of the sequence $\{T_{L_{p}}(p^{k})\}_{k}$; that is,

$$P_{L_{p}}(X)=\sum_{k\geq 0}T_{L_{p}}(p^{k})X^{k}.$$

Next, the formal Dirichlet series $\widehat{D}_{L}(s)$ and $D_{L}(s)$ are defined. We set

$$\widehat{T}_{L}(n)=\sum_{\widehat{\alpha}}\widehat{T}_{L}(\widehat{\alpha}),\quad T_{L}(n)=\sum_{\alpha}T_{L}(\alpha),$$

where $\widehat{\alpha}$ $(\text{resp.}\ \alpha)$ runs through a complete system of representatives of $\widehat{\Gamma}_{\!L}\backslash\widehat{\mathcal{A}}_{L}(n)/\widehat{\Gamma}_{\!L}$ $(\text{resp.}\ \Gamma_{\!L}\backslash\mathcal{A}_{L}(n)/\Gamma_{\!L})$. The formal Dirichlet series $\widehat{D}_{L}(s)$ and $D_{L}(s)$ are the generating functions of the sequences of $\{\widehat{T}_{L}(n)\}_{n}$ and $\{T_{L}(n)\}_{n}$, respectively; that is,

$$\widehat{D}_{L}(s)=\sum_{n>0}\widehat{T}_{L}(n)n^{-s},\quad D_{L}(s)=\sum_{n>0}T_{L}(n)n^{-s}.$$

Next, $P_{L_{p}}(X)$ is related to $\widehat{D}_{L}(s)$. For each element $\widehat{\alpha}$ of $\widehat{\Delta}_{L}$, let $\alpha_{p}$ denote its $\Delta_{L_{p}}$component. Then, $\widehat{T}_{L}(\widehat{\alpha})=\prod_{p}T_{L_{p}}(\alpha_{p})$ is obtained, where $p$ runs over all prime numbers. Here, this infinite product is meaningful since its terms commute with each other according to Proposition 2.1, and almost all of them are equal to $1$. Consequently, the following theorem is proven:

Finally, $\widehat{D}_{L}(s)$ is related to $D_{L}(s)$ using the additive map $\eta_{L}^{*}:\widehat{R}_{L}\to R_{L}$. It is evident that $\eta_{L}^{*}$ maps $\widehat{T}_{L}(n)$ to $T_{L}(n)$ for each positive integer $n$. Thus, the Euler product formula for $D_{L}(s)$ is proven.

Using the notations in Section 3, the theory of the Hecke ring with general linear groups as reported by Hecke [6], Shimura [17], and Tamagawa [18] is considered.

Let $r$ be a positive integer. Clearly, $G_{\mathbb{Z}^{r}}$ and $G_{{\mathbb{Z}^{r}_{p}}}$ are identified with $GL_{r}(\mathbb{Q})$ and $GL_{r}(\mathbb{Q}_{p})$, respectively. Similarly, we have $\Delta_{\mathbb{Z}^{r}}=M_{r}(\mathbb{Z})\cap GL_{r}(\mathbb{Q})$, $\Delta_{\mathbb{Z}^{r}_{p}}=M_{r}(\mathbb{Z}_{p})\cap GL_{r}(\mathbb{Q}_{p})$, $\Gamma_{\!\mathbb{Z}^{r}}=GL_{r}(\mathbb{Z})$, and, $\Gamma_{\!\mathbb{Z}^{r}_{p}}=GL_{r}(\mathbb{Z}_{p})$. Thus, the Hecke rings $R_{\mathbb{Z}^{r}}$ and $R_{\mathbb{Z}^{r}_{p}}$ coincide with the Hecke rings treated in [6], [17], and [18]. Furthermore, the Hecke ring $\widehat{R}_{\mathbb{Z}^{r}}$ is identified with $R_{\mathbb{Z}^{r}}$ as follows:

Certainly, the formal power series $P_{\mathbb{Z}^{r}_{p}}$(X) equals the local Hecke series treated in [6] and [18]. The following theorem was proved:

The series $D_{\mathbb{Z}^{r}}(s)$ is none other than the formal Dirichlet series treated in [17, Chapter 3]. Since $\eta_{*{\mathbb{Z}^{r}}}$ is bijective, the following theorem is obtained:

Therefore, the following theorem is obtained:

This section studies the proposed series in the case of the Heisenberg Lie algebra $\mathcal{H}$.