If you want the up/down keys to control the listbox then I think you need to manage the focus. Adding an explict call in your code as follows enables these keys.

foreach my $key ("a" , "b" , "c") {
    $lb -> insert( 'end', $key );
}
$lb->activate ( 0 ) ;
$lb->selectionSet ( 0 ) ;
$lb->focus();
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I'm _far_ from being a Tk expert, so I'd advise you to check out Tk::Focus docs for yourself.