Let \(G\) be a connected almost simple simply connected linear algebraic group over \(\mathbb Q\) with \(G(\mathbb R)\) non-compact. Let \(\Gamma\subset G( \mathbb Q)\) be a congruence subgroup. Let \(a\in G( \mathbb Q) \). For \(x\in\Gamma \setminus G(\mathbb R)\) set \(T_ax=\{[\Gamma a\Gamma x]\in \Gamma \setminus G(\mathbb R) \}\). Let \(T_a\) be the Hecke operator on \(L^2(\Gamma \setminus G(\mathbb R))\). Let \(L_0^2(\Gamma \setminus G(\mathbb R))\) be the orthogonal complement in \(L^2(\Gamma \setminus G(\mathbb R))\) of the subspace of constant functions, and let \(T_a^0: L_0^2(\Gamma \setminus G(\mathbb R))\to L_0^2(\Gamma \setminus G(\mathbb R))\) be the restriction of \(T_a\). The main goal of this paper is to present an \(L^2\)-norm estimate of \(T_a^0\) and, using that, to obtain an equidistribution of the sets \(T_ax\) as \(\text{deg}(a)\) tends to infinity with an estimate of equidistribution rate. For each finite prime \(p\) let \(A_p\) be a maximal \({\mathbb Q}_p\)-split torus of \(G({\mathbb Q}_p)\) and \(B_p\) a minimal \({\mathbb Q}_p\)-parabolic subgroup of \(G({\mathbb Q}_p)\) containing \(A_p\). Let \(\Phi_p\) be the set of non-multipliable roots in the relative \({\mathbb Q}_p\)-root system \(\Phi(G, A_p)\) with the ordering given by \(B_p\). Let \(K_p\) be a good maximal compact subgroup of \(G({\mathbb Q}_p)\). Let \(R_f\) be the set of finite primes. Theorem 1. Let \(R_1=\{p\in R_f\mid \text{rank}_{{\mathbb Q}_p}G=1 \}\) and \(R_2=\{p \in R_f\mid \text{rank}_{{\mathbb Q}_p}G\geq 2 \}\). There exists a constant \(C\) such that for any \(a\in G( \mathbb Q)\) \[ \| T_a^0 \| \leq C\left(\prod_{p\in R_1}\xi_{{\mathcal S}_p}^{1/2}(a) \right) \left(\prod_{p\in R_2}\xi_{{\mathcal S}_p}(a) \right)\,\text{if rank}_{\mathbb Q}G\geq 1 \] and \[ \| T_a^0 \| \leq C\left(\prod_{p\in R_2}\xi_{{\mathcal S}_p}(a) \right)\,\text{if rank}_{\mathbb Q}G=0, \] where \({\mathcal S}_p\) is a strongly orthogonal system of \(\Phi_p\) for each \(p\in R_1\cup R_2\) and \( \xi_{{\mathcal S}_p}\) is the bi-\(K_p\)-invariant function on \(G({\mathbb Q}_p)\). Theorem 2. If we assume that \(\text{rank}_{\mathbb Q}G\geq 1\) and that the Ramanujan conjecture for \(SL_2\) holds at every finite prime \(p\), then there exists a constant \(C\) such that for any \(a\in G( \mathbb Q) \) \[ \| T_a^0 \| \leq C\left(\prod_{p\in R_f}\xi_{{\mathcal S}_p}(a) \right). \] Theorem 3. Let \(G=SL_1(D)\) where \(D\) is a quaternion algebra over \(\mathbb Q \) such that \(D\) is split over \(\mathbb R\). Then there exists a constant \(C\) such that for any \(a\in G( \mathbb Q) \) \[ \| T_a^0 \| \leq C\prod_{p\in R_1}\xi_{{\mathcal S}_p}^{1/2}(a) \] where \({\mathcal S}_p\) is a strongly orthogonal system of \(\Phi_p\) for each \(p\in R_1\) ( in fact, \({\mathcal S}_p=\{\alpha_p\}\) for the simple root \(\alpha_p\) of \(\Phi_p\)). Corollary 4. Let \(G\) and \(\Gamma\) be as in Theorem 1. Let \(\{a_n\in G( \mathbb Q)\mid n\in {\mathbb N} \}\) be any sequence with \(\text{deg}(a_n)\to \infty\). If \(\text{rank}_{\mathbb Q}G=0\), assume that the diagonal embedding of the sequence \(\{a_n\}\) into the direct product \(\prod_{p\in R_2}G( {\mathbb Q}_p)\) is unbounded. Then \[ \lim_{n\to \infty} \| T_{a_n}^0 \|=0. \] Theorem 5. Let \(G=SL_n (n\geq 3)\) or \(Sp_{2n}(n\geq 2)\), and \(\Gamma=G(\mathbb Z) \). Let \({\mathcal S}\) be a maximal strongly orthogonal system of \(\Phi(G,A)\), where \(A\) is the diagonal subgroup of \(G\). Then there exists a function \(f\in L_0^2(\Gamma \setminus G(\mathbb R))\) with \(\| f \|=1\) such that for any \(\varepsilon>0\), one can find a constant \(C\) (depending on \(\varepsilon\)) such that \[ C \prod_{p\in R_f}\xi_{\mathcal S}^{1+\varepsilon}(a)\leq\| T_a^0f \|\leq \prod_{p\in R_f}\xi_{\mathcal S}(a) \] for any \(a\in A^+\). Theorem 6. Let \(G\), \( \Gamma\) and \(\{a_n\} \) be as in Corollary 4. Then for any \(x\in\Gamma \setminus G(\mathbb R)\), the sets \(T_{a_n}x\) are equidistributed with respect to \(d\mu\) ( the normalized Haar measure on \( \Gamma \setminus G(\mathbb R) \)), in the sense that \[ \lim_{n\to \infty}T_{a_n}f(x)=\int_{\Gamma \setminus G(\mathbb R)}f(g)d\mu(g) \] for any continuous function \(f\) on \( \Gamma \setminus G(\mathbb R)\) with compact support. Theorem 7. Keeping the same notation from Theorem 1, let \(f\) be a smooth function on \(\Gamma \setminus G(\mathbb R)\) with a compact support. Then for any \(a\in G( \mathbb Q)\) and for any \( x\in \Gamma \setminus G(\mathbb R)\), \[ \left | T_af(x)-\int_{\Gamma \setminus G(\mathbb R)}f(g)d\mu(g) \right|\leq C\left(\prod_{p\in R_1}\xi_{{\mathcal S}_p}^{1/2}(a) \right) \left(\prod_{p\in R_2}\xi_{{\mathcal S}_p}(a) \right)\,\text{if rank}_{\mathbb Q}G\geq 1 \] and \[ \left | T_af(x)-\int_{\Gamma \setminus G(\mathbb R)}f(g)d\mu(g) \right|\leq C\left(\prod_{p\in R_2}\xi_{{\mathcal S}_p}(a) \right)\,\text{if rank}_{\mathbb Q}G=0, \] where the constant \(C\) (depending on \(f\)) can be taken uniformly over compact subsets. The authors discuss some applications of the above results on equidistribution of lattices in \({\mathbb R}^n\). Let \(X\) be the space of equivalence classes of lattices where \(\Lambda\sim\Lambda'\) if and only if \(\Lambda=c\Lambda'\) for \(c\in{\mathbb R}^\times\). Let \(\overline{\Lambda}\) be the class in \(X\) represented by a lattice \(\Lambda\) in \({\mathbb R}^n\). For any \(n\) positive integers \(a_1,\ldots, a_n\) with \(a_{i+1}|a_i\) for all \(1\leq i \leq n-1\) and \(a_n=1\) let \[ X_{\overline{\Lambda}}(a_1,\ldots, a_n)=\{ \overline{\Lambda'}\mid \Lambda'\subset \Lambda, \Lambda\diagup\Lambda'\simeq\sum_{i=1}^{n-1}{\mathbb Z}\diagup a_i{\mathbb Z} \}. \] It holds that \[ X_{\overline{\Lambda}}=T_{diag(a_1,\ldots, a_n)}(\overline{\Lambda}). \] Corollary 8. (1) For any \( \overline{\Lambda}\in X\) the sets \(X_{\overline{\Lambda}}(a_1,\ldots, a_n)\) are equidistributed, that is, for any (nice) compact subset \(\Omega\subset X\) \[ \frac{|\Omega\cap X_{\overline{\Lambda}}(a_1,\ldots, a_n) |}{|X_{\overline{\Lambda}}(a_1,\ldots, a_n)|}\sim \mu(\Omega ) \] as \(diag(a_1,\ldots, a_n) \) goes to infinity in \(GL_n(\mathbb Z)\) modulo its center. (2) Let \(n \geq 3\). For any smooth function \(f\) on \(X\) with compact support, we have that for any \(\varepsilon>0 \), there exists a constant \(C\) (depending on \(\varepsilon \) and \(f\)) such that for any such \((a_1,\ldots, a_n)\) as above, \[ \left|\frac{\sum_{\overline{\Lambda'}\in X_{\overline{\Lambda}}(a_1,\ldots, a_n)}f(\overline{\Lambda'})}{|X_{\overline{\Lambda}}(a_1,\ldots, a_n)|}- \int_Xfd\mu\right|\leq C\prod_{i=1}^{[n/2]}\left(\frac{a_i}{a_{n+1-i}}\right)^{-1/2+\varepsilon} \] for any \(\overline{\Lambda}\in X.\) Corollary 9. (1) For any \( \overline{\Lambda}\in X\) and for any (nice) compact subset \( \Omega\subset X,\) \[ \frac{|\Omega\cap X_{\overline{\Lambda}}(m) |}{|X_{\overline{\Lambda}}(m)|}\sim \mu(\Omega ) \] as \(m\to\infty\), where \[ X_{\overline{\Lambda}}(m)=\{ \overline{\Lambda'}\mid \Lambda'\subset \Lambda, \det(\Lambda')=m\}. \] (2) Let \(n \geq 3\). Let \(f\) be a smooth function on \(X\) with compact support and \( \overline{\Lambda}\in X\). Then for any \(\varepsilon>0 \), we have \[ \frac{\sum_{\overline{\Lambda'}\in X_{\overline{\Lambda}}(m)}f(\overline{\Lambda'})}{|X_{\overline{\Lambda}}(m)|}= \int_Xfd\mu+O(m^{-1/2+\varepsilon}) \] as \(m\to\infty\).