Symplectic Steinberg group I have several questions about Steinberg group and K2 for symplectic group:


*

*Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?

*Does the center of symplectic Steinberg group coincide with K2 (the kernel of $\mathrm{SpSt}\rightarrow\mathrm{Sp}$ as usual)?

*Is there an analogue for Matsumoto's theorem?


I tryed to read "Sur les sous-groupes arithmetiques des groupes semi-simples deployes" by Hideya Matsumoto and all I got to know about symplectic case is that there is some problems with long roots in Cl. Also, is it written in english anywhere about non-Al K-theory? There is "The Classical groups and K-theory" by Hahn and O'Meara, but it tells about SL and about unitary groups only.
 A: Here's a small follow-up on Matsumoto's thesis, which deals essentially with
the congruence subgroup problem for Chevalley (split) algebraic groups.   This
followed work by Bass-Milnor-Serre, but in turn was followed by more technical
work on nonsplit groups (Prasad-Raghunathan, in particular).    In my 1980
Springer Lecture Notes 789 on Arithmetic Groups, I tried in the last part to
convey Matsumoto's ideas in the special case of $SL_n$ when $n \geq 3$ but
with some side remarks about the general case.  The congruence subgroup problem
has a different solution for $n=2$, which should for this and some other
purposes be assigned to type $C_1$ rather than the conventional $A_1$.  For
symplectic groups there is a significant difference, mentioned in my Remark on
page 129.  Briefly:
The ideas in my $SL_n$ proof carry over almost unchanged to other Chevalley groups
of rank $\geq 2$, with one important modification due to the fact that real Lie groups
of type $C$ starting with rank 1 have an infinite cyclic fundamental group while
others have a finite fundamental group.   This is what complicates the
proof for symplectic groups in Matsumoto's paper (which is only available in
French).    The special feature of root systems of type $C$ seems to be the
existence of roots equal to twice a weight. 
A: There is useful information about the symplectic analogues of the Steinberg group, the Steinberg symbols, and the $K_2$ functor in some of Michael Stein's papers from the '70's.  In particular, see his papers "Generators, relations and coverings of Chevalley groups over commutative rings" and "Surjective stability in dimension $0$ for $K_{2}$ and related functors" and "Injective stability for $K_{2}$ of local rings".
A: There is an excellent survey paper: Linear Algebraic Groups and K-theory http://users.ictp.it/~pub_off/lectures/lns023/Rehmann/Rehmann.pdf by Ulf Rehmann. It seems that Matsumoto paper concerns symplectic case, that is the answers to all your questions are positive. 
Note that for non-symplectic groups, Steinberg symbols are bilinear, however for symplectic it is not true. For a nice description of the 2-cocycle of the topological universal cover $\widetilde{SL_2({\mathbb R})}$ of $SL_2({\mathbb R})$ see e.g. Asai, T.: The reciprocity of Dedekind sums and the factor set for the universal covering group of $SL(2,{\mathbb R})$. 
From Asai work, one can deduce that the Steinberg Symbol corresponding to a $\widetilde{SL_2({\mathbb R})}$ is defined as: for $x,y\in {\mathbb R}^{\times}$ 
$c(x,y) = \left\{
  \begin{array}{l l}
    -1 & \quad \text{if } x < 0 \text{ and } y<0 \\
    0 & \quad \text{otherwise}\\
  \end{array} \right.$
A: Let me do a bit of necromancy here and address the third question.
A Note on Milnor–Witt K-Theory and a Theorem of Suslin by K. Hutchinson and L. Tao provides a description for $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ for an infinite field $F$ as Milnor—Witt K-theory $K_2^{MW}(F)$, introduced by F. Morel in 2003 in his study of $\mathbb{A}^1$-homotopy theory.
$K_*^{MW}(F)$ is a graded associative ring generated by the symbols $[u]$, $u\in F^*$ of degree $+1$ and one symbol $\eta$ of degree $-1$ modulo the following relations:


*

*For $a\in F\setminus\{0,1\}$, $[a]\cdot[1-a]=0$;

*For $a,b,\in F^*$, $[ab]=[a]+[b]+\eta[a][b]$;

*For $u\in F^*$, $[u]\eta=\eta[u]$;

*$\eta^2[-1]+2\eta=0$.


The proof is based on Matsumoto—Moore presentation for $H_2\left(Sp(F),\mathbb{Z}\right)$ and the coincidence of $K_2^{MW}(F)$ with $K_2^{MM}(F)$.
PS. The equality $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ has something to do with $\mathsf{A}_1=\mathsf{C}_1$ (see this MO question).
A: Recently, centrality and Suslin's local-global principle for symplectic K_2 are studied by Andrei Lavrenov, student of Nikolai Vavilov. 
