For a restricted Lie algebra \(\mathfrak g\) over an algebraically closed field \(K\) of characteristic \(p>0\) the elements \(x^p-x^{[p]}\), \(x\in \mathfrak g\), are contained in the center of the universal enveloping algebra \(U(\mathfrak g)\). Thus, the description of simple \(\mathfrak g\)-modules is confined to the description of simple \(U_{\chi }(\mathfrak g)\)-modules, where \(\chi \in \mathfrak g^*\) and \(U_{\chi }(\mathfrak g)\) is the quotient of \(U(\mathfrak g)\) by the ideal generated by \(x^p-x^{[p]}-\chi(x)^p\cdot 1\), \(x\in \mathfrak g\).
Now, let \(G\) be an almost simple and simply connected algebraic group. \(\mathfrak g=\text{Lie}(G)\), \(\mathfrak g=\mathfrak n^-+\mathfrak h+\mathfrak n^+\) is the triangular decomposition of \(\mathfrak g\). In case of good characteristic \(\mathfrak g\cong \mathfrak g^*\). This isomorphism allows to transfer Jordan decomposition to \(\mathfrak g^*\), \(\chi =\chi _s+\chi _n.\)
The general case was reduced to the case of nilpotent \(\chi \) by B. Yu. Weisfeiler and V. G. Kac [Funct. Anal. Appl. 5, 111-117 (1971; Zbl 0237.17003)] and E. M. Friedlander and B. J. Parshall [Am. J. Math. 110, 1055-1093 (1988; Zbl 0673.17010)]. Changing \(\chi \) by a conjugate one with respect to \(G\) we can consider that \(\chi (\mathfrak b)=0\), where \(\mathfrak b=\mathfrak h+\mathfrak n^+\). Therefore, each simple \(U_{\chi }(\mathfrak g)\)-module is covered by an induced module \(Z_{\chi }(\lambda)=U_{\chi }(\mathfrak g)\otimes _{U_{\chi }(\mathfrak b)}K_{\lambda }\) where \(K_{\lambda }\) is a one-dimensional \(\mathfrak b\)-module, \(\lambda \in \mathfrak h^*\), \(\mathfrak n^+K_{\lambda }=0.\)
If \(\chi\) is regular nilpotent (i.e. \(\dim \mathfrak c_{\mathfrak g}(\chi)=\dim \mathfrak h\)) then each \(Z_{\chi }(\lambda)\) is simple [see A. N. Panov, Funct. Anal. Appl. 23, 240-241 (1989); translation from Funkts. Anal. Prilozh. 23, No. 3, 80-81 (1989; Zbl 0688.17005), E. M. Friedlander and B. J. Parshall, Am. J. Math. 112, 375-395 (1990; Zbl 0714.17007)]. In the paper an important step has been taken in the study of \(Z_{\chi }(\lambda)\) where \(\chi \) is subregular nilpotent (i.e. \(\dim \mathfrak c_{\mathfrak g}(\chi)=\dim \mathfrak h + 2\)). The cases \(\mathfrak g=\mathfrak{sl}_{n+1}(K) ~(n+1\not \equiv 0(p))\) and \(\mathfrak {so}_{2n+1} ~(p\neq 2)\) have been treated. For these algebras there exist subregular nilpotents \(\chi \) which are of standard Levi form. This means that for some subset \(I\) of simple roots \(\chi (x_{-\alpha })\neq 0\) for \(\alpha \in I\) and \(\chi (x_{\alpha })=0\) for the rest of roots. When \(\chi \) has standard Levi form \(Z_{\chi }(\lambda)\) contains the unique maximal submodule. The corresponding simple quotient is denoted by \(L_{\chi }(\lambda)\). Thus, any simple \(U_{\chi }(\mathfrak g)\)-module is isomorphic to some \(L_{\chi }(\lambda)\). In case when \(\mathfrak g=\mathfrak{sl}_{n+1}\) the element \(e\) corresponding to \(\chi \) under the isomorphism \(\mathfrak g\cong \mathfrak g^*\) is chosen as \(e=\sum _{i=1}^{n-1}c_iE_{i,i+1}\), \(c_i\in K\), \(c_i\neq 0\) where \(E_{i,i+1}\) are standard matrix units and in case \(\mathfrak{so}_{2n+1}\), \(e=\sum _{i=2}^n c_ix_{\alpha _i}\) where \(\{ \alpha _1,\ldots ,\alpha _n\}\) is a standard basis of root system of type \(B_n\). For such subregular \(\chi \) it is proved that \(Z_{\chi }(\lambda)\) is uniserial and each composition factor has multiplicity 1. The complete description of all simple \(U_{\chi }(\mathfrak g)\)-modules \(L_{\chi }(\lambda)\) is given. Besides, the structure of \(Z_{\chi }(\lambda)\) as \(U_{\chi }(\mathfrak g)-T_0\)-modules is investigated, where \(T_0\) is the torus in \(G\), \(T_0=\bigcap _{i=1}^{n-1}\ker \alpha _i\) for \(\mathfrak{sl}_{n+1}\) and \(T_0=\bigcap _{i=2}^n\ker \alpha _i\) for \(\mathfrak{so}_{2n+1}\). The Ext groups for simple non-isomorphic modules are calculated.