We introduce a revised notion of gauge action in relation to Leavitt path algebras. This notion is based on the Laurent polynomial algebra and captures the full information of the grading on the algebra as it is the case of the gauge action of the graph $C^{\ast }$-algebra.

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,\dots ,x_6)$ via $k$-automorphisms defined by $\sigma \cdot x_i=x_{\sigma \left(i\right)}$ for any $\sigma \in G$ and any $1\le i\le 6$. We prove the following theorem. The fixed field $k(x_1,\dots ,x_6{)}^G$ is rational (i.e., purely transcendental) over $k$, except possibly when $G$ is isomorphic to ${PSL}_2\left({\mathbb{F}}_5\right)$, ${PGL}_2\left({\mathbb{F}}_5\right)$, or $A_6$. When $G$ is isomorphic to ${PSL}_2\left({\mathbb{F}}_5\right)$ or ${PGL}_2\left({\mathbb{F}}_5\right)$, then $\mathbb{C}(x_1,\dots ,x_6{)}^G$ is $\mathbb{C}$-rational and $k(x_1,\dots ,x_6{)}^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.

The behavior of orbits of the Hamilton vector field $H_p$ of the principal symbol $p$ of a second-order hyperbolic differential operator is discussed. In our previous paper, assuming that $p$ is noneffectively hyperbolic on the doubly characteristic manifold $\Sigma$ of $p$, we have proved that if $H_S^{3}p=0$ on $\Sigma$ with the Hamilton vector field $H_S$ of some specified $S$, then there exists a bicharacteristic landing on $\Sigma$ tangentially. The aim of this paper is to provide a much more simple proof of this result since the previous proof was fairly long and rather complicated.

Let $F$ be a non-archimedean local field. Recently, Broussous, Sécherre, and Stevens extended the notion of an endo-class, introduced by Bushnell and Henniart for ${GL}_N\left(F\right)$ with $N\ge 1$, to an inner form of ${GL}_N\left(F\right)$ over $F$, and conjectured that this endo-class for discrete series representations is preserved by the Jacquet–Langlands correspondence. Explicit realizations of the correspondence are given by Silberger and Zink for level-zero discrete series representations and by Bushnell and Henniart for totally ramified ones. In this paper, we show that these realizations confirm the conjecture.

Let $K=\mathbf{Q}\left(\sqrt{-D}\right)$ be an imaginary quadratic field with discriminant $-D$, and let $\chi$ be the Dirichlet character corresponding to the extension $K/\mathbf{Q}$. Let $m=2n$ or $2n+1$ with $n$ a positive integer. Let $f$ be a primitive form of weight $2k+1$ and character $\chi$ for ${\Gamma }_0\left(D\right)$ or a primitive form of weight $2k$ for ${SL}_2\left(\mathbf{Z}\right)$ according to whether $m=2n$ or $m=2n+1$. For such an $f$ let $I_m\left(f\right)$ be the lift of $f$ to the space of Hermitian modular forms constructed by Ikeda. We then give an explicit formula of the Koecher–Maass series $L(s,I_m(f\left)\right)$ of $I_m\left(f\right)$. This is a generalization of Mizuno.

We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on $2$-dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appear in the theory of Fourier–Mukai transforms. As an application, we generalize the Fourier–Mukai duality for K3 surfaces to our situation.

We describe prime ideals of height $2$ minimally generated by three elements in a Gorenstein, Nagata local ring of Krull dimension $3$ and multiplicity at most $3$. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.

In this paper, we introduce a motivic version of Toën’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of Toën and Kontsevich–Soibelman, respectively, are Drinfeld dual pairs, not only in the classical case (by counting over finite fields) but also in the motivic version. Consequently they are canonically isomorphic. All proofs, including that for the most important associative property, are deduced in a self-contained way by analyzing the symmetry properties around the octahedral axiom, a method we used previously.