It is possible to add and remove search fields on the advanced search page by editing the opac/parts/config.tt2 file in your template directory. Look for this section of the file:

search.adv_config = [
    {adv_label => l("Item Type"), adv_attr => ["mattype", "item_type"]},
    {adv_label => l("Item Form"), adv_attr => "item_form"},
    {adv_label => l("Language"),  adv_attr => "item_lang"},
    {adv_label => l("Audience"),  adv_attr => ["audience_group", "audience"], adv_break => 1},
    {adv_label => l("Video Format"), adv_attr => "vr_format"},
    {adv_label => l("Bib Level"), adv_attr => "bib_level"},
    {adv_label => l("Literary Form"), adv_attr => "lit_form", adv_break => 1},
    {adv_label => l("Search Library"), adv_special => "lib_selector"},
    {adv_label => l("Publication Year"), adv_special => "pub_year"},
    {adv_label => l("Sort Results"), adv_special => "sort_selector"},
];

For example, if you delete the line:

{adv_label => l("Language"),  adv_attr => "item_lang"},

the language field will no longer appear on your advanced search page. Changes will appear immediately after you save your changes.